| L(s) = 1 | − 1.82·2-s + 1.33·4-s + 0.563·5-s + 3.34·7-s + 1.21·8-s − 1.02·10-s − 6.11·11-s + 2.34·13-s − 6.11·14-s − 4.88·16-s − 6.16·17-s − 4.83·19-s + 0.753·20-s + 11.1·22-s + 2.50·23-s − 4.68·25-s − 4.28·26-s + 4.47·28-s + 5.48·29-s − 7.48·31-s + 6.50·32-s + 11.2·34-s + 1.88·35-s + 0.750·37-s + 8.83·38-s + 0.682·40-s − 4.56·41-s + ⋯ |
| L(s) = 1 | − 1.29·2-s + 0.668·4-s + 0.251·5-s + 1.26·7-s + 0.428·8-s − 0.325·10-s − 1.84·11-s + 0.650·13-s − 1.63·14-s − 1.22·16-s − 1.49·17-s − 1.11·19-s + 0.168·20-s + 2.38·22-s + 0.521·23-s − 0.936·25-s − 0.840·26-s + 0.845·28-s + 1.01·29-s − 1.34·31-s + 1.14·32-s + 1.93·34-s + 0.318·35-s + 0.123·37-s + 1.43·38-s + 0.107·40-s − 0.713·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.82T + 2T^{2} \) |
| 5 | \( 1 - 0.563T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 + 6.11T + 11T^{2} \) |
| 13 | \( 1 - 2.34T + 13T^{2} \) |
| 17 | \( 1 + 6.16T + 17T^{2} \) |
| 19 | \( 1 + 4.83T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 - 5.48T + 29T^{2} \) |
| 31 | \( 1 + 7.48T + 31T^{2} \) |
| 37 | \( 1 - 0.750T + 37T^{2} \) |
| 41 | \( 1 + 4.56T + 41T^{2} \) |
| 43 | \( 1 - 6.66T + 43T^{2} \) |
| 47 | \( 1 - 7.06T + 47T^{2} \) |
| 53 | \( 1 + 5.57T + 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 4.72T + 71T^{2} \) |
| 73 | \( 1 + 14.8T + 73T^{2} \) |
| 79 | \( 1 + 2.16T + 79T^{2} \) |
| 83 | \( 1 - 15.3T + 83T^{2} \) |
| 89 | \( 1 - 4.26T + 89T^{2} \) |
| 97 | \( 1 + 8.10T + 97T^{2} \) |
| 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
−9.182194508338962494649920435490, −8.588423979624353372999521508071, −7.962259163224980496960329476138, −7.33833044914632853224212428929, −6.16662949394865413157622259756, −5.03397735198901837534351323794, −4.33697843072874445068502507500, −2.48305926292531706556335012734, −1.66865126958768523449486237516, 0,
1.66865126958768523449486237516, 2.48305926292531706556335012734, 4.33697843072874445068502507500, 5.03397735198901837534351323794, 6.16662949394865413157622259756, 7.33833044914632853224212428929, 7.962259163224980496960329476138, 8.588423979624353372999521508071, 9.182194508338962494649920435490
