| L(s) = 1 | + 2-s + 1.82·3-s + 4-s − 2.37·5-s + 1.82·6-s + 8-s + 0.329·9-s − 2.37·10-s + 6.06·11-s + 1.82·12-s − 5.90·13-s − 4.33·15-s + 16-s − 6.48·17-s + 0.329·18-s − 7.02·19-s − 2.37·20-s + 6.06·22-s − 3.36·23-s + 1.82·24-s + 0.641·25-s − 5.90·26-s − 4.87·27-s − 1.41·29-s − 4.33·30-s + 6.55·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.05·3-s + 0.5·4-s − 1.06·5-s + 0.744·6-s + 0.353·8-s + 0.109·9-s − 0.751·10-s + 1.82·11-s + 0.526·12-s − 1.63·13-s − 1.11·15-s + 0.250·16-s − 1.57·17-s + 0.0777·18-s − 1.61·19-s − 0.531·20-s + 1.29·22-s − 0.702·23-s + 0.372·24-s + 0.128·25-s − 1.15·26-s − 0.937·27-s − 0.262·29-s − 0.791·30-s + 1.17·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 1.82T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 11 | \( 1 - 6.06T + 11T^{2} \) |
| 13 | \( 1 + 5.90T + 13T^{2} \) |
| 17 | \( 1 + 6.48T + 17T^{2} \) |
| 19 | \( 1 + 7.02T + 19T^{2} \) |
| 23 | \( 1 + 3.36T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 6.55T + 31T^{2} \) |
| 37 | \( 1 + 4.18T + 37T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 - 7.43T + 47T^{2} \) |
| 53 | \( 1 + 1.96T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 8.10T + 61T^{2} \) |
| 67 | \( 1 + 8.19T + 67T^{2} \) |
| 71 | \( 1 - 7.71T + 71T^{2} \) |
| 73 | \( 1 - 1.99T + 73T^{2} \) |
| 79 | \( 1 + 2.63T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 5.10T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.199444502156731586141664731369, −7.20907854478233504046123246657, −6.79330967387132489288612061406, −5.95573776162164539768657524269, −4.54510791124451039900112361239, −4.26026098067333554992630380247, −3.58594620773942025022765370843, −2.55334014951595465838135440757, −1.91594800338131051482075868024, 0,
1.91594800338131051482075868024, 2.55334014951595465838135440757, 3.58594620773942025022765370843, 4.26026098067333554992630380247, 4.54510791124451039900112361239, 5.95573776162164539768657524269, 6.79330967387132489288612061406, 7.20907854478233504046123246657, 8.199444502156731586141664731369
