| L(s) = 1 | − 2·2-s + 2·4-s + 5-s + 5·7-s − 2·10-s − 2·11-s − 10·14-s − 4·16-s − 2·17-s + 2·20-s + 4·22-s − 6·23-s + 25-s + 10·28-s + 4·29-s − 7·31-s + 8·32-s + 4·34-s + 5·35-s − 2·37-s − 6·41-s + 43-s − 4·44-s + 12·46-s + 8·47-s + 18·49-s − 2·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s + 1.88·7-s − 0.632·10-s − 0.603·11-s − 2.67·14-s − 16-s − 0.485·17-s + 0.447·20-s + 0.852·22-s − 1.25·23-s + 1/5·25-s + 1.88·28-s + 0.742·29-s − 1.25·31-s + 1.41·32-s + 0.685·34-s + 0.845·35-s − 0.328·37-s − 0.937·41-s + 0.152·43-s − 0.603·44-s + 1.76·46-s + 1.16·47-s + 18/7·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−7.63191830349450506511065357207, −7.32424451922789077347092107222, −6.27120901501207590227932710169, −5.43210262742652919238429528027, −4.76304990812899016071365029911, −4.09272495499005085504981921440, −2.61280199081632125119869153918, −1.86315542563040252055657950846, −1.34475697320078698804067361472, 0,
1.34475697320078698804067361472, 1.86315542563040252055657950846, 2.61280199081632125119869153918, 4.09272495499005085504981921440, 4.76304990812899016071365029911, 5.43210262742652919238429528027, 6.27120901501207590227932710169, 7.32424451922789077347092107222, 7.63191830349450506511065357207
