| L(s) = 1 | − 2-s + 4-s − 3.64·5-s + 2·7-s − 8-s + 3.64·10-s + 3.64·11-s − 5.29·13-s − 2·14-s + 16-s + 7.29·17-s + 5.64·19-s − 3.64·20-s − 3.64·22-s + 23-s + 8.29·25-s + 5.29·26-s + 2·28-s − 1.29·29-s + 9.29·31-s − 32-s − 7.29·34-s − 7.29·35-s − 8.93·37-s − 5.64·38-s + 3.64·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.63·5-s + 0.755·7-s − 0.353·8-s + 1.15·10-s + 1.09·11-s − 1.46·13-s − 0.534·14-s + 0.250·16-s + 1.76·17-s + 1.29·19-s − 0.815·20-s − 0.777·22-s + 0.208·23-s + 1.65·25-s + 1.03·26-s + 0.377·28-s − 0.239·29-s + 1.66·31-s − 0.176·32-s − 1.25·34-s − 1.23·35-s − 1.46·37-s − 0.915·38-s + 0.576·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8515062077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8515062077\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3.64T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 - 7.29T + 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 29 | \( 1 + 1.29T + 29T^{2} \) |
| 31 | \( 1 - 9.29T + 31T^{2} \) |
| 37 | \( 1 + 8.93T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 5.64T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 3.64T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.64T + 61T^{2} \) |
| 67 | \( 1 - 0.937T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 8.35T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 9.29T + 97T^{2} \) |
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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
−11.42040563994266866501743132035, −10.26095027768618246511897003410, −9.378722773826353719771765495979, −8.287703697821199902486276496520, −7.59310279482573095032487397250, −7.07016137321053242716383940365, −5.37637725804369239791115941708, −4.22026374078375503376121968005, −3.05176169464901095133350979583, −1.03252784750203758561527897719,
1.03252784750203758561527897719, 3.05176169464901095133350979583, 4.22026374078375503376121968005, 5.37637725804369239791115941708, 7.07016137321053242716383940365, 7.59310279482573095032487397250, 8.287703697821199902486276496520, 9.378722773826353719771765495979, 10.26095027768618246511897003410, 11.42040563994266866501743132035
