| L(s) = 1 | − 56·7-s + 104·9-s + 80·17-s − 128·23-s + 672·25-s + 48·31-s − 32·41-s + 592·47-s + 1.76e3·49-s − 5.82e3·63-s + 1.16e3·71-s − 1.53e3·73-s + 1.55e3·79-s + 4.43e3·81-s + 5.15e3·89-s + 3.28e3·97-s + 752·103-s + 1.56e3·113-s − 4.48e3·119-s + 3.69e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8.32e3·153-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 3.85·9-s + 1.14·17-s − 1.16·23-s + 5.37·25-s + 0.278·31-s − 0.121·41-s + 1.83·47-s + 36/7·49-s − 11.6·63-s + 1.95·71-s − 2.46·73-s + 2.21·79-s + 6.07·81-s + 6.13·89-s + 3.43·97-s + 0.719·103-s + 1.30·113-s − 3.45·119-s + 2.77·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 4.39·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.378338385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.378338385\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( ( 1 + p T )^{8} \) |
| good | 3 | \( 1 - 104 T^{2} + 2128 p T^{4} - 271208 T^{6} + 8402398 T^{8} - 271208 p^{6} T^{10} + 2128 p^{13} T^{12} - 104 p^{18} T^{14} + p^{24} T^{16} \) |
| 5 | \( 1 - 672 T^{2} + 1808 p^{3} T^{4} - 9702576 p T^{6} + 7207600734 T^{8} - 9702576 p^{7} T^{10} + 1808 p^{15} T^{12} - 672 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( 1 - 336 p T^{2} + 7886332 T^{4} - 12242902032 T^{6} + 16244782978086 T^{8} - 12242902032 p^{6} T^{10} + 7886332 p^{12} T^{12} - 336 p^{19} T^{14} + p^{24} T^{16} \) |
| 13 | \( 1 - 10800 T^{2} + 60966992 T^{4} - 222586320480 T^{6} + 576266222349726 T^{8} - 222586320480 p^{6} T^{10} + 60966992 p^{12} T^{12} - 10800 p^{18} T^{14} + p^{24} T^{16} \) |
| 17 | \( ( 1 - 40 T + 732 p T^{2} - 634008 T^{3} + 77656582 T^{4} - 634008 p^{3} T^{5} + 732 p^{7} T^{6} - 40 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 19 | \( 1 - 20376 T^{2} + 10763984 p T^{4} - 1209062702904 T^{6} + 6977079111266142 T^{8} - 1209062702904 p^{6} T^{10} + 10763984 p^{13} T^{12} - 20376 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( ( 1 + 64 T + 42000 T^{2} + 2010432 T^{3} + 723262078 T^{4} + 2010432 p^{3} T^{5} + 42000 p^{6} T^{6} + 64 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 29 | \( 1 - 85256 T^{2} + 3121262396 T^{4} - 69040659720312 T^{6} + 1421702540707727846 T^{8} - 69040659720312 p^{6} T^{10} + 3121262396 p^{12} T^{12} - 85256 p^{18} T^{14} + p^{24} T^{16} \) |
| 31 | \( ( 1 - 24 T + 36044 T^{2} - 4119480 T^{3} + 1278352614 T^{4} - 4119480 p^{3} T^{5} + 36044 p^{6} T^{6} - 24 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 - 255688 T^{2} + 29856798844 T^{4} - 2197301022202168 T^{6} + \)\(12\!\cdots\!62\)\( T^{8} - 2197301022202168 p^{6} T^{10} + 29856798844 p^{12} T^{12} - 255688 p^{18} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 + 16 T + 105276 T^{2} - 26099856 T^{3} + 4672195558 T^{4} - 26099856 p^{3} T^{5} + 105276 p^{6} T^{6} + 16 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 - 312432 T^{2} + 51310967036 T^{4} - 6155146092780048 T^{6} + \)\(56\!\cdots\!90\)\( T^{8} - 6155146092780048 p^{6} T^{10} + 51310967036 p^{12} T^{12} - 312432 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( ( 1 - 296 T + 361692 T^{2} - 73372680 T^{3} + 52799197702 T^{4} - 73372680 p^{3} T^{5} + 361692 p^{6} T^{6} - 296 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 53 | \( 1 - 635848 T^{2} + 214634579644 T^{4} - 48116434944800824 T^{6} + \)\(81\!\cdots\!54\)\( T^{8} - 48116434944800824 p^{6} T^{10} + 214634579644 p^{12} T^{12} - 635848 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( 1 - 711000 T^{2} + 299452153840 T^{4} - 87478655645921400 T^{6} + \)\(20\!\cdots\!62\)\( T^{8} - 87478655645921400 p^{6} T^{10} + 299452153840 p^{12} T^{12} - 711000 p^{18} T^{14} + p^{24} T^{16} \) |
| 61 | \( 1 - 1171264 T^{2} + 704615447056 T^{4} - 271626771638144848 T^{6} + \)\(73\!\cdots\!14\)\( T^{8} - 271626771638144848 p^{6} T^{10} + 704615447056 p^{12} T^{12} - 1171264 p^{18} T^{14} + p^{24} T^{16} \) |
| 67 | \( 1 - 1154128 T^{2} + 606079663036 T^{4} - 214746424141490992 T^{6} + \)\(66\!\cdots\!70\)\( T^{8} - 214746424141490992 p^{6} T^{10} + 606079663036 p^{12} T^{12} - 1154128 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 - 584 T + 542460 T^{2} + 14444184 T^{3} + 67443285286 T^{4} + 14444184 p^{3} T^{5} + 542460 p^{6} T^{6} - 584 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( ( 1 + 768 T + 947308 T^{2} + 808867008 T^{3} + 437175361158 T^{4} + 808867008 p^{3} T^{5} + 947308 p^{6} T^{6} + 768 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 79 | \( ( 1 - 776 T + 1891676 T^{2} - 1136400744 T^{3} + 1380231878534 T^{4} - 1136400744 p^{3} T^{5} + 1891676 p^{6} T^{6} - 776 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 3185672 T^{2} + 4996880472752 T^{4} - 4963094153913351624 T^{6} + \)\(33\!\cdots\!86\)\( T^{8} - 4963094153913351624 p^{6} T^{10} + 4996880472752 p^{12} T^{12} - 3185672 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 - 2576 T + 4515324 T^{2} - 5400247152 T^{3} + 5161182144166 T^{4} - 5400247152 p^{3} T^{5} + 4515324 p^{6} T^{6} - 2576 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( ( 1 - 1640 T + 39340 p T^{2} - 4338255896 T^{3} + 5279068197670 T^{4} - 4338255896 p^{3} T^{5} + 39340 p^{7} T^{6} - 1640 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.55627588773991541081478507135, −4.16192246598551399590451063122, −4.15365499869222478461822973687, −3.77678854716165811999525560967, −3.72989327202727491797549037099, −3.63279870875002694659537199832, −3.61362209683993799492084741927, −3.55913808465975228687567324088, −3.18452658509830826173691254243, −3.09751629320462774234136684740, −3.00629632686919538229464159218, −2.74684832997680286099030341416, −2.63201860602163337706816025988, −2.29357338680693274262117579953, −2.11378031176429819767271728647, −2.09581166357654148543924546717, −2.08152805246567388171186709130, −1.47234196829049855737354522447, −1.28770277008948365800325638183, −1.09419210668911998996346875130, −1.09386186249239053576628043528, −0.77933432229732956495676147117, −0.76161153510839818893709501890, −0.54352675323946321305358273371, −0.13930166607260991448520496947,
0.13930166607260991448520496947, 0.54352675323946321305358273371, 0.76161153510839818893709501890, 0.77933432229732956495676147117, 1.09386186249239053576628043528, 1.09419210668911998996346875130, 1.28770277008948365800325638183, 1.47234196829049855737354522447, 2.08152805246567388171186709130, 2.09581166357654148543924546717, 2.11378031176429819767271728647, 2.29357338680693274262117579953, 2.63201860602163337706816025988, 2.74684832997680286099030341416, 3.00629632686919538229464159218, 3.09751629320462774234136684740, 3.18452658509830826173691254243, 3.55913808465975228687567324088, 3.61362209683993799492084741927, 3.63279870875002694659537199832, 3.72989327202727491797549037099, 3.77678854716165811999525560967, 4.15365499869222478461822973687, 4.16192246598551399590451063122, 4.55627588773991541081478507135
Plot not available for L-functions of degree greater than 10.
