| L(s) = 1 | − 728·5-s + 3.68e3·9-s − 1.26e4·13-s − 3.91e5·17-s − 9.11e5·25-s − 7.05e5·29-s + 4.44e6·37-s + 2.95e6·41-s − 2.68e6·45-s + 1.29e7·49-s − 4.50e6·53-s − 4.01e7·61-s + 9.19e6·65-s − 5.92e6·73-s − 2.41e7·81-s + 2.85e8·85-s + 1.32e8·89-s + 5.64e7·97-s − 2.31e8·101-s + 5.57e8·109-s − 4.99e8·113-s − 4.65e7·117-s + 6.21e8·121-s + 9.52e8·125-s + ⋯ |
| L(s) = 1 | − 1.16·5-s + 0.561·9-s − 0.442·13-s − 4.69·17-s − 2.33·25-s − 0.997·29-s + 2.37·37-s + 1.04·41-s − 0.654·45-s + 2.24·49-s − 0.570·53-s − 2.90·61-s + 0.515·65-s − 0.208·73-s − 0.561·81-s + 5.46·85-s + 2.11·89-s + 0.637·97-s − 2.22·101-s + 3.94·109-s − 3.06·113-s − 0.248·117-s + 2.90·121-s + 3.90·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6965481493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6965481493\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 1228 p T^{2} + 4192150 p^{2} T^{4} - 1228 p^{17} T^{6} + p^{32} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 + 364 T + 130926 p T^{2} + 364 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12949124 T^{2} + 2124073815270 p^{2} T^{4} - 12949124 p^{16} T^{6} + p^{32} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 621783908 T^{2} + 181770025950389574 T^{4} - 621783908 p^{16} T^{6} + p^{32} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 6316 T - 471179994 T^{2} + 6316 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 195996 T + 21760239302 T^{2} + 195996 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 10037112676 T^{2} + \)\(45\!\cdots\!06\)\( T^{4} - 10037112676 p^{16} T^{6} + p^{32} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 230553122948 T^{2} + \)\(25\!\cdots\!54\)\( T^{4} - 230553122948 p^{16} T^{6} + p^{32} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 352748 T + 999396063654 T^{2} + 352748 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1561001128964 T^{2} + \)\(12\!\cdots\!10\)\( p^{2} T^{4} - 1561001128964 p^{16} T^{6} + p^{32} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 2221524 T + 8217364493222 T^{2} - 2221524 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 1476676 T + 2025162183942 T^{2} - 1476676 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 9479410588004 T^{2} + \)\(25\!\cdots\!90\)\( T^{4} - 9479410588004 p^{16} T^{6} + p^{32} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 32334216312836 T^{2} + \)\(95\!\cdots\!66\)\( T^{4} - 32334216312836 p^{16} T^{6} + p^{32} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 2250924 T + 123936144680870 T^{2} + 2250924 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 305254750575716 T^{2} + \)\(59\!\cdots\!46\)\( T^{4} - 305254750575716 p^{16} T^{6} + p^{32} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 20079532 T + 283245594359334 T^{2} + 20079532 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 1540404309460580 T^{2} + \)\(92\!\cdots\!98\)\( T^{4} - 1540404309460580 p^{16} T^{6} + p^{32} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 783735682299524 T^{2} + \)\(23\!\cdots\!50\)\( T^{4} - 783735682299524 p^{16} T^{6} + p^{32} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 2960412 T + 1505171737656134 T^{2} + 2960412 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 4755170211804164 T^{2} + \)\(98\!\cdots\!70\)\( T^{4} - 4755170211804164 p^{16} T^{6} + p^{32} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 5209133236219748 T^{2} + \)\(14\!\cdots\!74\)\( T^{4} - 5209133236219748 p^{16} T^{6} + p^{32} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 66323940 T + 6939976370286662 T^{2} - 66323940 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 28222436 T + 13755128655195846 T^{2} - 28222436 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.79311460963032039208021045444, −10.76248687954902712011943446728, −10.06697530192770405285114237441, −9.429371226119558077918527099296, −9.400648019126051485386470637534, −9.180300247733256859018160395106, −8.502570113137408173272631281172, −8.462325418541455206241791374352, −7.60628965205222927206051476597, −7.57351118866909932846050249272, −7.41715028560302545998238239567, −6.64209756391555686733626621510, −6.49194101977894172402773192351, −5.93537970653782596092755562246, −5.67833114888102104889711452203, −4.65739849250976241896225651397, −4.41460824133692604749422474687, −4.25647132429671964340172265985, −3.98413711600608749817441039965, −3.18369814818584980981364185652, −2.35812650589682538521024542821, −2.19684909296229404503866106389, −1.72129451089016326807996198709, −0.57250205771830771567202340033, −0.25328177459807282278534473988,
0.25328177459807282278534473988, 0.57250205771830771567202340033, 1.72129451089016326807996198709, 2.19684909296229404503866106389, 2.35812650589682538521024542821, 3.18369814818584980981364185652, 3.98413711600608749817441039965, 4.25647132429671964340172265985, 4.41460824133692604749422474687, 4.65739849250976241896225651397, 5.67833114888102104889711452203, 5.93537970653782596092755562246, 6.49194101977894172402773192351, 6.64209756391555686733626621510, 7.41715028560302545998238239567, 7.57351118866909932846050249272, 7.60628965205222927206051476597, 8.462325418541455206241791374352, 8.502570113137408173272631281172, 9.180300247733256859018160395106, 9.400648019126051485386470637534, 9.429371226119558077918527099296, 10.06697530192770405285114237441, 10.76248687954902712011943446728, 10.79311460963032039208021045444
