| L(s) = 1 | − 2.69·2-s + 5.24·4-s + 5-s + 1.69·7-s − 8.74·8-s − 2.69·10-s + 4.60·11-s − 4.55·14-s + 13.0·16-s − 5.38·17-s − 4.96·19-s + 5.24·20-s − 12.3·22-s − 6.76·23-s + 25-s + 8.87·28-s − 3.64·29-s + 5.59·31-s − 17.6·32-s + 14.4·34-s + 1.69·35-s + 3.86·37-s + 13.3·38-s − 8.74·40-s + 10.8·41-s − 9.91·43-s + 24.1·44-s + ⋯ |
| L(s) = 1 | − 1.90·2-s + 2.62·4-s + 0.447·5-s + 0.639·7-s − 3.09·8-s − 0.851·10-s + 1.38·11-s − 1.21·14-s + 3.25·16-s − 1.30·17-s − 1.13·19-s + 1.17·20-s − 2.64·22-s − 1.41·23-s + 0.200·25-s + 1.67·28-s − 0.676·29-s + 1.00·31-s − 3.11·32-s + 2.48·34-s + 0.286·35-s + 0.635·37-s + 2.16·38-s − 1.38·40-s + 1.70·41-s − 1.51·43-s + 3.64·44-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 | \( 1 + 2.69T + 2T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 + 4.96T + 19T^{2} \) |
| 23 | \( 1 + 6.76T + 23T^{2} \) |
| 29 | \( 1 + 3.64T + 29T^{2} \) |
| 31 | \( 1 - 5.59T + 31T^{2} \) |
| 37 | \( 1 - 3.86T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 9.91T + 43T^{2} \) |
| 47 | \( 1 + 4.80T + 47T^{2} \) |
| 53 | \( 1 - 9.99T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 7.44T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 5.03T + 71T^{2} \) |
| 73 | \( 1 + 0.917T + 73T^{2} \) |
| 79 | \( 1 + 5.11T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 + 3.85T + 89T^{2} \) |
| 97 | \( 1 + 5.27T + 97T^{2} \) |
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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
−7.77566840770975412777323509436, −6.94281768928344149055958837701, −6.33422352700186524468381900865, −6.02102610696569499660407652580, −4.63978313881497832163514193035, −3.83442200245848607974886249874, −2.51244125426976013765003863354, −1.93930295517690778426521293765, −1.23928442521590381211279681186, 0,
1.23928442521590381211279681186, 1.93930295517690778426521293765, 2.51244125426976013765003863354, 3.83442200245848607974886249874, 4.63978313881497832163514193035, 6.02102610696569499660407652580, 6.33422352700186524468381900865, 6.94281768928344149055958837701, 7.77566840770975412777323509436
