| L(s) = 1 | + 2·4-s − 3·9-s + 8·11-s + 16-s + 20·19-s + 8·29-s + 16·31-s − 6·36-s + 8·41-s + 16·44-s + 14·49-s + 50·59-s − 4·61-s − 2·64-s − 32·71-s + 40·76-s − 28·79-s + 16·81-s + 14·89-s − 24·99-s − 40·101-s − 24·109-s + 16·116-s − 52·121-s + 32·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 4-s − 9-s + 2.41·11-s + 1/4·16-s + 4.58·19-s + 1.48·29-s + 2.87·31-s − 36-s + 1.24·41-s + 2.41·44-s + 2·49-s + 6.50·59-s − 0.512·61-s − 1/4·64-s − 3.79·71-s + 4.58·76-s − 3.15·79-s + 16/9·81-s + 1.48·89-s − 2.41·99-s − 3.98·101-s − 2.29·109-s + 1.48·116-s − 4.72·121-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.83765726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.83765726\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| good | 3 | \( 1 + p T^{2} - 7 T^{4} - 2 p T^{6} + 94 T^{8} - 2 p^{3} T^{10} - 7 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} \) |
| 7 | \( ( 1 - p T^{2} + 4 T^{4} - p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - T + p T^{2} )^{8} \) |
| 13 | \( ( 1 - T^{2} + p^{2} T^{4} )^{2}( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | \( 1 + 32 T^{2} + 258 T^{4} + 6016 T^{6} + 207299 T^{8} + 6016 p^{2} T^{10} + 258 p^{4} T^{12} + 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 + 11 T^{2} - 623 T^{4} - 3454 T^{6} + 209686 T^{8} - 3454 p^{2} T^{10} - 623 p^{4} T^{12} + 11 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - 67 T^{2} + 3516 T^{4} - 67 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 - 4 T - 53 T^{2} + 52 T^{3} + 2424 T^{4} + 52 p T^{5} - 53 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 136 T^{2} + 10242 T^{4} + 619616 T^{6} + 30510611 T^{8} + 619616 p^{2} T^{10} + 10242 p^{4} T^{12} + 136 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 + 56 T^{2} + 1266 T^{4} - 142688 T^{6} - 8310205 T^{8} - 142688 p^{2} T^{10} + 1266 p^{4} T^{12} + 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 + 143 T^{2} + 10233 T^{4} + 657514 T^{6} + 39034934 T^{8} + 657514 p^{2} T^{10} + 10233 p^{4} T^{12} + 143 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 25 T + 355 T^{2} - 3800 T^{3} + 32544 T^{4} - 3800 p T^{5} + 355 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 2 T - 102 T^{2} - 32 T^{3} + 7271 T^{4} - 32 p T^{5} - 102 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 - 5 T^{2} - 6711 T^{4} + 11210 T^{6} + 25110350 T^{8} + 11210 p^{2} T^{10} - 6711 p^{4} T^{12} - 5 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 16 T + 118 T^{2} - 64 T^{3} - 4173 T^{4} - 64 p T^{5} + 118 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 175 T^{2} + 15409 T^{4} + 797650 T^{6} + 39927790 T^{8} + 797650 p^{2} T^{10} + 15409 p^{4} T^{12} + 175 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( ( 1 + 14 T + 6 T^{2} + 448 T^{3} + 14375 T^{4} + 448 p T^{5} + 6 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 - 119 T^{2} + 17212 T^{4} - 119 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 7 T - 103 T^{2} + 182 T^{3} + 10822 T^{4} + 182 p T^{5} - 103 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + 271 T^{2} + 39361 T^{4} + 4136002 T^{6} + 381910750 T^{8} + 4136002 p^{2} T^{10} + 39361 p^{4} T^{12} + 271 p^{6} T^{14} + p^{8} T^{16} \) |
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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
−4.19373740451360005703593306738, −4.15055269490509394462503077305, −4.03398890125590011690737397796, −3.95562363345995292918732389383, −3.80323521017333767440659696986, −3.63136913140628961018503893458, −3.61374450524243981506358456389, −3.14228494469975446328887367978, −3.13327750597882642498346383346, −3.07891629297677710230774717574, −2.97001578291980625893395780918, −2.80649472652805392429985314933, −2.79792616949914753569088012668, −2.50038256090244495007386632541, −2.38838964476666400701231461527, −2.26042249001501973880793771821, −2.07666962745830228321772681539, −1.83752064659743478617453500853, −1.53222304177150500109006465160, −1.29059717817252957577298194546, −1.08445201268065939575002623277, −1.02515014743117884489911891253, −1.02112526799805556563942386418, −0.912765476266462427843701852331, −0.32072699169141836731112303232,
0.32072699169141836731112303232, 0.912765476266462427843701852331, 1.02112526799805556563942386418, 1.02515014743117884489911891253, 1.08445201268065939575002623277, 1.29059717817252957577298194546, 1.53222304177150500109006465160, 1.83752064659743478617453500853, 2.07666962745830228321772681539, 2.26042249001501973880793771821, 2.38838964476666400701231461527, 2.50038256090244495007386632541, 2.79792616949914753569088012668, 2.80649472652805392429985314933, 2.97001578291980625893395780918, 3.07891629297677710230774717574, 3.13327750597882642498346383346, 3.14228494469975446328887367978, 3.61374450524243981506358456389, 3.63136913140628961018503893458, 3.80323521017333767440659696986, 3.95562363345995292918732389383, 4.03398890125590011690737397796, 4.15055269490509394462503077305, 4.19373740451360005703593306738
Plot not available for L-functions of degree greater than 10.
