| L(s) = 1 | − 3-s + 5-s + 2·7-s − 2·9-s − 11-s − 4·13-s − 15-s − 2·17-s − 2·21-s + 23-s − 4·25-s + 5·27-s + 7·31-s + 33-s + 2·35-s − 3·37-s + 4·39-s + 8·41-s + 6·43-s − 2·45-s − 8·47-s − 3·49-s + 2·51-s + 6·53-s − 55-s + 5·59-s + 12·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s − 2/3·9-s − 0.301·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s − 0.436·21-s + 0.208·23-s − 4/5·25-s + 0.962·27-s + 1.25·31-s + 0.174·33-s + 0.338·35-s − 0.493·37-s + 0.640·39-s + 1.24·41-s + 0.914·43-s − 0.298·45-s − 1.16·47-s − 3/7·49-s + 0.280·51-s + 0.824·53-s − 0.134·55-s + 0.650·59-s + 1.53·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.41610144939628, −14.16764119392301, −13.41507138687767, −13.09157455870674, −12.26594723344191, −12.04211498344849, −11.35026719609234, −11.13021123897663, −10.48798819216079, −9.863009852790048, −9.609009369348891, −8.679246521369799, −8.449863588203762, −7.715640093865206, −7.267678390659656, −6.558702182842392, −6.050336872333391, −5.456604810551564, −5.044739700362480, −4.524299022348822, −3.868370309107363, −2.794414378650488, −2.497388762858484, −1.742979068183725, −0.8365829788754063, 0,
0.8365829788754063, 1.742979068183725, 2.497388762858484, 2.794414378650488, 3.868370309107363, 4.524299022348822, 5.044739700362480, 5.456604810551564, 6.050336872333391, 6.558702182842392, 7.267678390659656, 7.715640093865206, 8.449863588203762, 8.679246521369799, 9.609009369348891, 9.863009852790048, 10.48798819216079, 11.13021123897663, 11.35026719609234, 12.04211498344849, 12.26594723344191, 13.09157455870674, 13.41507138687767, 14.16764119392301, 14.41610144939628
