| L(s) = 1 | − 2.55·2-s − 3-s + 4.51·4-s + 5-s + 2.55·6-s + 4.92·7-s − 6.41·8-s + 9-s − 2.55·10-s − 0.347·11-s − 4.51·12-s − 2.97·13-s − 12.5·14-s − 15-s + 7.34·16-s + 6.49·17-s − 2.55·18-s + 6.17·19-s + 4.51·20-s − 4.92·21-s + 0.886·22-s + 6.41·24-s + 25-s + 7.60·26-s − 27-s + 22.2·28-s − 4.05·29-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.577·3-s + 2.25·4-s + 0.447·5-s + 1.04·6-s + 1.86·7-s − 2.26·8-s + 0.333·9-s − 0.807·10-s − 0.104·11-s − 1.30·12-s − 0.825·13-s − 3.36·14-s − 0.258·15-s + 1.83·16-s + 1.57·17-s − 0.601·18-s + 1.41·19-s + 1.00·20-s − 1.07·21-s + 0.188·22-s + 1.30·24-s + 0.200·25-s + 1.49·26-s − 0.192·27-s + 4.20·28-s − 0.752·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.140054269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140054269\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 7 | \( 1 - 4.92T + 7T^{2} \) |
| 11 | \( 1 + 0.347T + 11T^{2} \) |
| 13 | \( 1 + 2.97T + 13T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 - 6.17T + 19T^{2} \) |
| 29 | \( 1 + 4.05T + 29T^{2} \) |
| 31 | \( 1 - 7.03T + 31T^{2} \) |
| 37 | \( 1 + 2.11T + 37T^{2} \) |
| 41 | \( 1 + 3.94T + 41T^{2} \) |
| 43 | \( 1 - 3.20T + 43T^{2} \) |
| 47 | \( 1 - 5.11T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 + 7.32T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 0.401T + 71T^{2} \) |
| 73 | \( 1 + 6.69T + 73T^{2} \) |
| 79 | \( 1 - 8.07T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.903514815175218320295576388551, −7.43662054083822664993075937035, −6.85549274037579664407176962172, −5.70552635703833835153535950956, −5.35160983437761314188472463942, −4.53247314930668373395256918705, −3.12676555380262022724851492006, −2.14894484020308125628830587499, −1.41248343031844932257636023916, −0.812718483378291639102890405644,
0.812718483378291639102890405644, 1.41248343031844932257636023916, 2.14894484020308125628830587499, 3.12676555380262022724851492006, 4.53247314930668373395256918705, 5.35160983437761314188472463942, 5.70552635703833835153535950956, 6.85549274037579664407176962172, 7.43662054083822664993075937035, 7.903514815175218320295576388551
