| L(s) = 1 | + 0.423·2-s − 1.82·4-s − 3.15·5-s − 2.93·7-s − 1.61·8-s − 1.33·10-s + 0.303·11-s − 2.46·13-s − 1.24·14-s + 2.95·16-s + 0.539·17-s + 19-s + 5.75·20-s + 0.128·22-s − 5.82·23-s + 4.98·25-s − 1.04·26-s + 5.34·28-s + 8.39·29-s + 2.68·31-s + 4.49·32-s + 0.228·34-s + 9.27·35-s − 1.96·37-s + 0.423·38-s + 5.11·40-s + 2.99·41-s + ⋯ |
| L(s) = 1 | + 0.299·2-s − 0.910·4-s − 1.41·5-s − 1.10·7-s − 0.572·8-s − 0.423·10-s + 0.0915·11-s − 0.683·13-s − 0.332·14-s + 0.738·16-s + 0.130·17-s + 0.229·19-s + 1.28·20-s + 0.0274·22-s − 1.21·23-s + 0.996·25-s − 0.204·26-s + 1.00·28-s + 1.55·29-s + 0.482·31-s + 0.793·32-s + 0.0392·34-s + 1.56·35-s − 0.323·37-s + 0.0687·38-s + 0.808·40-s + 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.423T + 2T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 - 0.303T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 0.539T + 17T^{2} \) |
| 23 | \( 1 + 5.82T + 23T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 - 2.68T + 31T^{2} \) |
| 37 | \( 1 + 1.96T + 37T^{2} \) |
| 41 | \( 1 - 2.99T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 - 0.592T + 59T^{2} \) |
| 61 | \( 1 - 4.40T + 61T^{2} \) |
| 67 | \( 1 + 8.48T + 67T^{2} \) |
| 71 | \( 1 - 4.36T + 71T^{2} \) |
| 73 | \( 1 - 0.0473T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 6.09T + 83T^{2} \) |
| 89 | \( 1 - 8.02T + 89T^{2} \) |
| 97 | \( 1 + 7.95T + 97T^{2} \) |
| 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
−7.62619919100829165806680841026, −6.77358942283485351526389829827, −6.11584666317656404149698914251, −5.26459003102323908578745610405, −4.43928949975054489069640057638, −3.98959203941247100149721322888, −3.30707559799960621832671668481, −2.59266520781471254299553975101, −0.823504728045048826011813398706, 0,
0.823504728045048826011813398706, 2.59266520781471254299553975101, 3.30707559799960621832671668481, 3.98959203941247100149721322888, 4.43928949975054489069640057638, 5.26459003102323908578745610405, 6.11584666317656404149698914251, 6.77358942283485351526389829827, 7.62619919100829165806680841026
