| L(s) = 1 | − 2.82·2-s − 2.39·3-s + 5.96·4-s − 5-s + 6.74·6-s + 0.875·7-s − 11.1·8-s + 2.71·9-s + 2.82·10-s − 3.86·11-s − 14.2·12-s − 0.582·13-s − 2.47·14-s + 2.39·15-s + 19.6·16-s − 2.93·17-s − 7.67·18-s + 2.08·19-s − 5.96·20-s − 2.09·21-s + 10.8·22-s + 0.349·23-s + 26.7·24-s + 25-s + 1.64·26-s + 0.673·27-s + 5.22·28-s + ⋯ |
| L(s) = 1 | − 1.99·2-s − 1.38·3-s + 2.98·4-s − 0.447·5-s + 2.75·6-s + 0.330·7-s − 3.95·8-s + 0.906·9-s + 0.892·10-s − 1.16·11-s − 4.11·12-s − 0.161·13-s − 0.660·14-s + 0.617·15-s + 4.91·16-s − 0.712·17-s − 1.80·18-s + 0.478·19-s − 1.33·20-s − 0.456·21-s + 2.32·22-s + 0.0728·23-s + 5.46·24-s + 0.200·25-s + 0.322·26-s + 0.129·27-s + 0.987·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1831493269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1831493269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 2.82T + 2T^{2} \) |
| 3 | \( 1 + 2.39T + 3T^{2} \) |
| 7 | \( 1 - 0.875T + 7T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 + 0.582T + 13T^{2} \) |
| 17 | \( 1 + 2.93T + 17T^{2} \) |
| 19 | \( 1 - 2.08T + 19T^{2} \) |
| 23 | \( 1 - 0.349T + 23T^{2} \) |
| 29 | \( 1 - 3.11T + 29T^{2} \) |
| 31 | \( 1 + 4.04T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 7.84T + 43T^{2} \) |
| 47 | \( 1 - 5.82T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 - 1.51T + 59T^{2} \) |
| 61 | \( 1 + 4.55T + 61T^{2} \) |
| 67 | \( 1 + 3.14T + 67T^{2} \) |
| 71 | \( 1 - 5.72T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 7.49T + 89T^{2} \) |
| 97 | \( 1 - 3.87T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.998920908510352502441608243147, −8.816988681503182489914351244203, −8.239330080726271477270561163434, −7.29095788369338965338395444568, −6.80693373333733717575886261495, −5.79371913309827964195869815770, −5.00541925233618560712149911913, −3.12188345153097920378974242854, −1.82462837116061625224415256581, −0.44387074567230066065231266189,
0.44387074567230066065231266189, 1.82462837116061625224415256581, 3.12188345153097920378974242854, 5.00541925233618560712149911913, 5.79371913309827964195869815770, 6.80693373333733717575886261495, 7.29095788369338965338395444568, 8.239330080726271477270561163434, 8.816988681503182489914351244203, 9.998920908510352502441608243147
