| L(s) = 1 | − 2-s − 1.21·3-s + 4-s − 5-s + 1.21·6-s + 0.441·7-s − 8-s − 1.53·9-s + 10-s + 0.781·11-s − 1.21·12-s − 4.67·13-s − 0.441·14-s + 1.21·15-s + 16-s + 0.347·17-s + 1.53·18-s + 0.947·19-s − 20-s − 0.535·21-s − 0.781·22-s − 2.26·23-s + 1.21·24-s + 25-s + 4.67·26-s + 5.49·27-s + 0.441·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.699·3-s + 0.5·4-s − 0.447·5-s + 0.494·6-s + 0.166·7-s − 0.353·8-s − 0.510·9-s + 0.316·10-s + 0.235·11-s − 0.349·12-s − 1.29·13-s − 0.118·14-s + 0.312·15-s + 0.250·16-s + 0.0842·17-s + 0.360·18-s + 0.217·19-s − 0.223·20-s − 0.116·21-s − 0.166·22-s − 0.472·23-s + 0.247·24-s + 0.200·25-s + 0.917·26-s + 1.05·27-s + 0.0834·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 1.21T + 3T^{2} \) |
| 7 | \( 1 - 0.441T + 7T^{2} \) |
| 11 | \( 1 - 0.781T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 - 0.347T + 17T^{2} \) |
| 19 | \( 1 - 0.947T + 19T^{2} \) |
| 23 | \( 1 + 2.26T + 23T^{2} \) |
| 29 | \( 1 - 8.38T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 - 7.03T + 37T^{2} \) |
| 41 | \( 1 + 4.97T + 41T^{2} \) |
| 43 | \( 1 - 1.53T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 + 9.21T + 53T^{2} \) |
| 59 | \( 1 - 7.20T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 9.64T + 71T^{2} \) |
| 73 | \( 1 + 5.47T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 - 8.36T + 83T^{2} \) |
| 89 | \( 1 - 7.27T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
| 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
−8.045533610488998238142213431031, −7.51162835869472312073400172410, −6.53844693463555279023383155803, −6.12717093610571066232062788788, −4.99845170560557356198967057118, −4.54166235374781992509736239443, −3.17864461228497707040937586598, −2.44441774550427368580940787900, −1.06694878469586337632051625557, 0,
1.06694878469586337632051625557, 2.44441774550427368580940787900, 3.17864461228497707040937586598, 4.54166235374781992509736239443, 4.99845170560557356198967057118, 6.12717093610571066232062788788, 6.53844693463555279023383155803, 7.51162835869472312073400172410, 8.045533610488998238142213431031
