| L(s) = 1 | − 5-s − 4.08·7-s − 1.35·11-s + 0.648·13-s + 1.35·17-s − 0.648·19-s + 4.79·23-s + 25-s − 3.87·29-s + 7.69·31-s + 4.08·35-s + 7.52·37-s + 0.179·41-s + 0.820·43-s + 10.9·47-s + 9.69·49-s − 4.17·53-s + 1.35·55-s + 4.17·59-s − 3.82·61-s − 0.648·65-s − 8.14·67-s − 6.11·71-s − 12.3·73-s + 5.52·77-s − 10.3·79-s − 12.2·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.54·7-s − 0.407·11-s + 0.179·13-s + 0.327·17-s − 0.148·19-s + 0.998·23-s + 0.200·25-s − 0.719·29-s + 1.38·31-s + 0.690·35-s + 1.23·37-s + 0.0280·41-s + 0.125·43-s + 1.59·47-s + 1.38·49-s − 0.573·53-s + 0.182·55-s + 0.543·59-s − 0.489·61-s − 0.0803·65-s − 0.994·67-s − 0.725·71-s − 1.44·73-s + 0.629·77-s − 1.16·79-s − 1.34·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 4.08T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 - 0.648T + 13T^{2} \) |
| 17 | \( 1 - 1.35T + 17T^{2} \) |
| 19 | \( 1 + 0.648T + 19T^{2} \) |
| 23 | \( 1 - 4.79T + 23T^{2} \) |
| 29 | \( 1 + 3.87T + 29T^{2} \) |
| 31 | \( 1 - 7.69T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 - 0.179T + 41T^{2} \) |
| 43 | \( 1 - 0.820T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 4.17T + 53T^{2} \) |
| 59 | \( 1 - 4.17T + 59T^{2} \) |
| 61 | \( 1 + 3.82T + 61T^{2} \) |
| 67 | \( 1 + 8.14T + 67T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−7.47785512742683006992360613089, −7.08228639982128892468357321391, −6.16407248175384743239703205247, −5.76430405630096030805539800220, −4.66761811968449645659407121789, −3.96769684593637343680203096301, −3.07171836728557975181155403650, −2.64694511983250850789739797118, −1.09930225402076442174696554141, 0,
1.09930225402076442174696554141, 2.64694511983250850789739797118, 3.07171836728557975181155403650, 3.96769684593637343680203096301, 4.66761811968449645659407121789, 5.76430405630096030805539800220, 6.16407248175384743239703205247, 7.08228639982128892468357321391, 7.47785512742683006992360613089
