| L(s) = 1 | − 2-s − 1.68·3-s + 4-s + 3.48·5-s + 1.68·6-s − 7-s − 8-s − 0.176·9-s − 3.48·10-s + 3.39·11-s − 1.68·12-s + 0.689·13-s + 14-s − 5.86·15-s + 16-s + 2.88·17-s + 0.176·18-s + 3.48·20-s + 1.68·21-s − 3.39·22-s + 4.65·23-s + 1.68·24-s + 7.17·25-s − 0.689·26-s + 5.33·27-s − 28-s + 1.60·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.970·3-s + 0.5·4-s + 1.56·5-s + 0.685·6-s − 0.377·7-s − 0.353·8-s − 0.0589·9-s − 1.10·10-s + 1.02·11-s − 0.485·12-s + 0.191·13-s + 0.267·14-s − 1.51·15-s + 0.250·16-s + 0.700·17-s + 0.0416·18-s + 0.780·20-s + 0.366·21-s − 0.724·22-s + 0.970·23-s + 0.342·24-s + 1.43·25-s − 0.135·26-s + 1.02·27-s − 0.188·28-s + 0.298·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.400284794\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.400284794\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 1.68T + 3T^{2} \) |
| 5 | \( 1 - 3.48T + 5T^{2} \) |
| 11 | \( 1 - 3.39T + 11T^{2} \) |
| 13 | \( 1 - 0.689T + 13T^{2} \) |
| 17 | \( 1 - 2.88T + 17T^{2} \) |
| 23 | \( 1 - 4.65T + 23T^{2} \) |
| 29 | \( 1 - 1.60T + 29T^{2} \) |
| 31 | \( 1 + 7.34T + 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 - 5.13T + 41T^{2} \) |
| 43 | \( 1 + 7.95T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 + 0.941T + 53T^{2} \) |
| 59 | \( 1 - 12.1T + 59T^{2} \) |
| 61 | \( 1 + 7.54T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 9.42T + 73T^{2} \) |
| 79 | \( 1 - 6.77T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 - 6.89T + 97T^{2} \) |
| 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
−8.478774996486283562702746781633, −7.28994851234948631787864536778, −6.65948485530727466367319435835, −6.08167379106427063599003203163, −5.62126265573053613856383945708, −4.89043868448963394626765126215, −3.57876182360760065338997012821, −2.62530221028357569260927628674, −1.60149364642700474965469432247, −0.802599348137998310973912981613,
0.802599348137998310973912981613, 1.60149364642700474965469432247, 2.62530221028357569260927628674, 3.57876182360760065338997012821, 4.89043868448963394626765126215, 5.62126265573053613856383945708, 6.08167379106427063599003203163, 6.65948485530727466367319435835, 7.28994851234948631787864536778, 8.478774996486283562702746781633
