| L(s) = 1 | + 1.67e7·2-s − 4.81e11·3-s + 2.81e14·4-s − 9.77e16·5-s − 8.07e18·6-s − 1.91e20·7-s + 4.72e21·8-s − 7.78e21·9-s − 1.63e24·10-s + 2.53e25·11-s − 1.35e26·12-s − 1.03e27·13-s − 3.21e27·14-s + 4.70e28·15-s + 7.92e28·16-s + 8.07e29·17-s − 1.30e29·18-s − 3.81e31·19-s − 2.75e31·20-s + 9.21e31·21-s + 4.26e32·22-s + 1.80e33·23-s − 2.27e33·24-s − 8.21e33·25-s − 1.74e34·26-s + 1.18e35·27-s − 5.39e34·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.983·3-s + 0.5·4-s − 0.733·5-s − 0.695·6-s − 0.377·7-s + 0.353·8-s − 0.0325·9-s − 0.518·10-s + 0.777·11-s − 0.491·12-s − 0.530·13-s − 0.267·14-s + 0.721·15-s + 0.250·16-s + 0.576·17-s − 0.0230·18-s − 1.78·19-s − 0.366·20-s + 0.371·21-s + 0.549·22-s + 0.784·23-s − 0.347·24-s − 0.462·25-s − 0.375·26-s + 1.01·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(50-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+49/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(25)\) |
\(\approx\) |
\(0.8628396972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8628396972\) |
| \(L(\frac{51}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.67e7T \) |
| 7 | \( 1 + 1.91e20T \) |
| good | 3 | \( 1 + 4.81e11T + 2.39e23T^{2} \) |
| 5 | \( 1 + 9.77e16T + 1.77e34T^{2} \) |
| 11 | \( 1 - 2.53e25T + 1.06e51T^{2} \) |
| 13 | \( 1 + 1.03e27T + 3.83e54T^{2} \) |
| 17 | \( 1 - 8.07e29T + 1.95e60T^{2} \) |
| 19 | \( 1 + 3.81e31T + 4.55e62T^{2} \) |
| 23 | \( 1 - 1.80e33T + 5.30e66T^{2} \) |
| 29 | \( 1 + 1.04e36T + 4.54e71T^{2} \) |
| 31 | \( 1 - 2.15e36T + 1.19e73T^{2} \) |
| 37 | \( 1 + 4.78e38T + 6.94e76T^{2} \) |
| 41 | \( 1 + 2.69e38T + 1.06e79T^{2} \) |
| 43 | \( 1 + 1.37e40T + 1.09e80T^{2} \) |
| 47 | \( 1 + 6.92e40T + 8.56e81T^{2} \) |
| 53 | \( 1 - 1.63e42T + 3.08e84T^{2} \) |
| 59 | \( 1 + 2.25e43T + 5.91e86T^{2} \) |
| 61 | \( 1 - 5.72e43T + 3.02e87T^{2} \) |
| 67 | \( 1 - 7.90e43T + 3.00e89T^{2} \) |
| 71 | \( 1 + 4.06e45T + 5.14e90T^{2} \) |
| 73 | \( 1 - 4.94e45T + 2.00e91T^{2} \) |
| 79 | \( 1 + 6.03e46T + 9.63e92T^{2} \) |
| 83 | \( 1 + 2.77e46T + 1.08e94T^{2} \) |
| 89 | \( 1 - 1.26e46T + 3.31e95T^{2} \) |
| 97 | \( 1 + 4.45e48T + 2.24e97T^{2} \) |
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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
−11.36610803357109562606675586292, −10.24688444875485666452905142038, −8.621550166580617839651840570498, −7.15283714525956097977179538758, −6.28809571151742885145360098407, −5.25281118060909158197256999763, −4.19879830932058351566376318980, −3.23518046144624203389066997807, −1.77786963773334805215347719723, −0.35243240421970505006579977209,
0.35243240421970505006579977209, 1.77786963773334805215347719723, 3.23518046144624203389066997807, 4.19879830932058351566376318980, 5.25281118060909158197256999763, 6.28809571151742885145360098407, 7.15283714525956097977179538758, 8.621550166580617839651840570498, 10.24688444875485666452905142038, 11.36610803357109562606675586292
