| L(s) = 1 | − 2.29·3-s + 2.83·5-s − 4.51·7-s + 2.28·9-s − 5.79·11-s + 0.740·13-s − 6.50·15-s − 5.42·17-s + 10.3·21-s − 3.94·23-s + 3.01·25-s + 1.64·27-s − 5.50·29-s + 5.13·31-s + 13.3·33-s − 12.7·35-s + 6.55·37-s − 1.70·39-s − 6.98·41-s + 5.53·43-s + 6.46·45-s + 0.994·47-s + 13.3·49-s + 12.4·51-s + 1.00·53-s − 16.4·55-s + 0.364·59-s + ⋯ |
| L(s) = 1 | − 1.32·3-s + 1.26·5-s − 1.70·7-s + 0.760·9-s − 1.74·11-s + 0.205·13-s − 1.68·15-s − 1.31·17-s + 2.26·21-s − 0.823·23-s + 0.603·25-s + 0.317·27-s − 1.02·29-s + 0.923·31-s + 2.31·33-s − 2.15·35-s + 1.07·37-s − 0.272·39-s − 1.09·41-s + 0.844·43-s + 0.963·45-s + 0.145·47-s + 1.90·49-s + 1.74·51-s + 0.137·53-s − 2.21·55-s + 0.0474·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5232801557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5232801557\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.29T + 3T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 + 4.51T + 7T^{2} \) |
| 11 | \( 1 + 5.79T + 11T^{2} \) |
| 13 | \( 1 - 0.740T + 13T^{2} \) |
| 17 | \( 1 + 5.42T + 17T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 - 5.13T + 31T^{2} \) |
| 37 | \( 1 - 6.55T + 37T^{2} \) |
| 41 | \( 1 + 6.98T + 41T^{2} \) |
| 43 | \( 1 - 5.53T + 43T^{2} \) |
| 47 | \( 1 - 0.994T + 47T^{2} \) |
| 53 | \( 1 - 1.00T + 53T^{2} \) |
| 59 | \( 1 - 0.364T + 59T^{2} \) |
| 61 | \( 1 - 1.82T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 3.46T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 - 6.19T + 79T^{2} \) |
| 83 | \( 1 + 2.62T + 83T^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 + 0.874T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.013954470174067706365170575390, −7.942983908237343800158790203026, −6.80491515938533496422999492068, −6.38378974759673878892223716281, −5.72343767940540964902386069380, −5.29651067929328791301669730589, −4.20399499439474149170107384347, −2.88865402562044063757349037555, −2.17218109573658863277235516895, −0.44195226567705230577551452762,
0.44195226567705230577551452762, 2.17218109573658863277235516895, 2.88865402562044063757349037555, 4.20399499439474149170107384347, 5.29651067929328791301669730589, 5.72343767940540964902386069380, 6.38378974759673878892223716281, 6.80491515938533496422999492068, 7.942983908237343800158790203026, 9.013954470174067706365170575390
