| L(s) = 1 | + 0.235·2-s + 2.48·3-s − 1.94·4-s − 0.468·5-s + 0.583·6-s − 5.07·7-s − 0.928·8-s + 3.15·9-s − 0.110·10-s − 2.07·11-s − 4.82·12-s − 2.16·13-s − 1.19·14-s − 1.16·15-s + 3.67·16-s + 17-s + 0.742·18-s + 0.772·19-s + 0.910·20-s − 12.5·21-s − 0.487·22-s − 6.12·23-s − 2.30·24-s − 4.78·25-s − 0.509·26-s + 0.381·27-s + 9.86·28-s + ⋯ |
| L(s) = 1 | + 0.166·2-s + 1.43·3-s − 0.972·4-s − 0.209·5-s + 0.238·6-s − 1.91·7-s − 0.328·8-s + 1.05·9-s − 0.0348·10-s − 0.624·11-s − 1.39·12-s − 0.600·13-s − 0.319·14-s − 0.299·15-s + 0.917·16-s + 0.242·17-s + 0.174·18-s + 0.177·19-s + 0.203·20-s − 2.74·21-s − 0.103·22-s − 1.27·23-s − 0.470·24-s − 0.956·25-s − 0.0999·26-s + 0.0733·27-s + 1.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 0.235T + 2T^{2} \) |
| 3 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 + 0.468T + 5T^{2} \) |
| 7 | \( 1 + 5.07T + 7T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 + 2.16T + 13T^{2} \) |
| 19 | \( 1 - 0.772T + 19T^{2} \) |
| 23 | \( 1 + 6.12T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 31 | \( 1 - 0.668T + 31T^{2} \) |
| 37 | \( 1 + 0.278T + 37T^{2} \) |
| 41 | \( 1 + 8.44T + 41T^{2} \) |
| 47 | \( 1 - 0.605T + 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 + 3.15T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 + 0.0451T + 73T^{2} \) |
| 79 | \( 1 - 0.717T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 1.50T + 89T^{2} \) |
| 97 | \( 1 - 7.17T + 97T^{2} \) |
| 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
−9.801142288312741513053977528487, −9.220056300054697855390130570333, −8.290677630728566168049762839694, −7.64713741631871110274807315958, −6.48117352209499549879428233803, −5.37957511246547369688814931916, −3.97690512997423506447697250279, −3.41225045183356327721604889720, −2.48345684969225387867957189166, 0,
2.48345684969225387867957189166, 3.41225045183356327721604889720, 3.97690512997423506447697250279, 5.37957511246547369688814931916, 6.48117352209499549879428233803, 7.64713741631871110274807315958, 8.290677630728566168049762839694, 9.220056300054697855390130570333, 9.801142288312741513053977528487
