| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 12-s − 13-s − 2·14-s + 15-s + 16-s − 4·17-s − 18-s + 19-s − 20-s − 2·21-s + 24-s + 25-s + 26-s − 27-s + 2·28-s − 29-s − 30-s + 8·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s − 0.436·21-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 0.377·28-s − 0.185·29-s − 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.188033469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.188033469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−14.82611066068336, −14.09692656999871, −13.52197848749044, −13.01939487811806, −12.25221907544087, −11.91367751648432, −11.34460653743498, −11.12817132840395, −10.36073276841188, −10.07882435818027, −9.300123125778805, −8.784090930799833, −8.282411997998053, −7.629465433729753, −7.354872060845986, −6.487044316824967, −6.245060296659745, −5.323638595882300, −4.809071015956921, −4.282151816039678, −3.544392944535204, −2.608196096735856, −2.068006387270065, −1.151920219103826, −0.5090037390705537,
0.5090037390705537, 1.151920219103826, 2.068006387270065, 2.608196096735856, 3.544392944535204, 4.282151816039678, 4.809071015956921, 5.323638595882300, 6.245060296659745, 6.487044316824967, 7.354872060845986, 7.629465433729753, 8.282411997998053, 8.784090930799833, 9.300123125778805, 10.07882435818027, 10.36073276841188, 11.12817132840395, 11.34460653743498, 11.91367751648432, 12.25221907544087, 13.01939487811806, 13.52197848749044, 14.09692656999871, 14.82611066068336
