| L(s) = 1 | − 0.554·2-s − 1.69·4-s − 5-s − 4.49·7-s + 2.04·8-s + 0.554·10-s + 2·11-s + 2.49·14-s + 2.24·16-s − 2.91·17-s + 2.04·19-s + 1.69·20-s − 1.10·22-s − 4.35·23-s + 25-s + 7.60·28-s + 0.713·29-s − 5.89·31-s − 5.34·32-s + 1.61·34-s + 4.49·35-s + 11.2·37-s − 1.13·38-s − 2.04·40-s − 4.21·41-s − 2.59·43-s − 3.38·44-s + ⋯ |
| L(s) = 1 | − 0.392·2-s − 0.846·4-s − 0.447·5-s − 1.69·7-s + 0.724·8-s + 0.175·10-s + 0.603·11-s + 0.666·14-s + 0.561·16-s − 0.706·17-s + 0.470·19-s + 0.378·20-s − 0.236·22-s − 0.908·23-s + 0.200·25-s + 1.43·28-s + 0.132·29-s − 1.05·31-s − 0.944·32-s + 0.277·34-s + 0.759·35-s + 1.84·37-s − 0.184·38-s − 0.323·40-s − 0.659·41-s − 0.395·43-s − 0.510·44-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 | \( 1 + 0.554T + 2T^{2} \) |
| 7 | \( 1 + 4.49T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 19 | \( 1 - 2.04T + 19T^{2} \) |
| 23 | \( 1 + 4.35T + 23T^{2} \) |
| 29 | \( 1 - 0.713T + 29T^{2} \) |
| 31 | \( 1 + 5.89T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 4.21T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 8.71T + 53T^{2} \) |
| 59 | \( 1 + 11.9T + 59T^{2} \) |
| 61 | \( 1 - 1.36T + 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 + 1.78T + 89T^{2} \) |
| 97 | \( 1 - 1.20T + 97T^{2} \) |
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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
−7.65681441379992481072958395738, −6.80226912183172419039445887255, −6.28688966118754789963467546140, −5.48226524605998965330908279729, −4.53455383232306617704878689163, −3.82022614216430328827103559387, −3.38017826762730826814706021788, −2.25153873845371743684868633283, −0.881983954359822952811069600818, 0,
0.881983954359822952811069600818, 2.25153873845371743684868633283, 3.38017826762730826814706021788, 3.82022614216430328827103559387, 4.53455383232306617704878689163, 5.48226524605998965330908279729, 6.28688966118754789963467546140, 6.80226912183172419039445887255, 7.65681441379992481072958395738
