| L(s) = 1 | − 4·4-s − 4·5-s − 140·11-s + 16·16-s − 48·19-s + 16·20-s − 109·25-s + 432·29-s + 416·31-s + 412·41-s + 560·44-s + 682·49-s + 560·55-s − 740·59-s − 1.10e3·61-s − 64·64-s + 1.08e3·71-s + 192·76-s − 1.58e3·79-s − 64·80-s − 1.87e3·89-s + 192·95-s + 436·100-s + 1.18e3·101-s − 740·109-s − 1.72e3·116-s + 1.20e4·121-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.357·5-s − 3.83·11-s + 1/4·16-s − 0.579·19-s + 0.178·20-s − 0.871·25-s + 2.76·29-s + 2.41·31-s + 1.56·41-s + 1.91·44-s + 1.98·49-s + 1.37·55-s − 1.63·59-s − 2.30·61-s − 1/8·64-s + 1.80·71-s + 0.289·76-s − 2.25·79-s − 0.0894·80-s − 2.23·89-s + 0.207·95-s + 0.435·100-s + 1.16·101-s − 0.650·109-s − 1.38·116-s + 9.04·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7910493204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7910493204\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + 4 T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 682 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 1478 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9342 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 14334 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 216 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 208 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 206 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 73750 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 105246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 136150 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 370 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 550 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 71542 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 540 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 413218 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 792 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 980358 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 938 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1822210 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.10781147973330493133395981977, −13.44013595304228410489311100526, −12.83794671659031800713798951246, −12.35502594416933827554039524549, −11.99168109912117979018073388800, −10.78724315461149715897162670577, −10.76332721886548661014504211752, −10.10642314008639750098482114518, −9.778562641608067821768298416545, −8.578051630475317594238774977570, −8.250181407692054391244660353010, −7.83827582872334273059172737378, −7.28779139547374256604122749654, −6.14531695116063254432991272069, −5.61540191644989400971545842340, −4.67076659534695145022232315746, −4.52716288594949031469371517418, −2.79227728644389770554895895844, −2.68573485585703268549144005533, −0.51702683540305336395139394427,
0.51702683540305336395139394427, 2.68573485585703268549144005533, 2.79227728644389770554895895844, 4.52716288594949031469371517418, 4.67076659534695145022232315746, 5.61540191644989400971545842340, 6.14531695116063254432991272069, 7.28779139547374256604122749654, 7.83827582872334273059172737378, 8.250181407692054391244660353010, 8.578051630475317594238774977570, 9.778562641608067821768298416545, 10.10642314008639750098482114518, 10.76332721886548661014504211752, 10.78724315461149715897162670577, 11.99168109912117979018073388800, 12.35502594416933827554039524549, 12.83794671659031800713798951246, 13.44013595304228410489311100526, 14.10781147973330493133395981977
