| L(s) = 1 | + 0.193·2-s − 1.96·4-s + 0.806·5-s + 0.324·7-s − 0.768·8-s + 0.156·10-s + 4.63·11-s + 2.96·13-s + 0.0630·14-s + 3.77·16-s + 2·17-s + 6.63·19-s − 1.58·20-s + 0.899·22-s − 8.31·23-s − 4.35·25-s + 0.574·26-s − 0.637·28-s + 5.35·29-s + 5.61·31-s + 2.26·32-s + 0.387·34-s + 0.261·35-s − 2.41·37-s + 1.28·38-s − 0.619·40-s − 41-s + ⋯ |
| L(s) = 1 | + 0.137·2-s − 0.981·4-s + 0.360·5-s + 0.122·7-s − 0.271·8-s + 0.0494·10-s + 1.39·11-s + 0.821·13-s + 0.0168·14-s + 0.943·16-s + 0.485·17-s + 1.52·19-s − 0.353·20-s + 0.191·22-s − 1.73·23-s − 0.870·25-s + 0.112·26-s − 0.120·28-s + 0.993·29-s + 1.00·31-s + 0.401·32-s + 0.0665·34-s + 0.0442·35-s − 0.397·37-s + 0.208·38-s − 0.0979·40-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.370946108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.370946108\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 - 0.193T + 2T^{2} \) |
| 5 | \( 1 - 0.806T + 5T^{2} \) |
| 7 | \( 1 - 0.324T + 7T^{2} \) |
| 11 | \( 1 - 4.63T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 6.63T + 19T^{2} \) |
| 23 | \( 1 + 8.31T + 23T^{2} \) |
| 29 | \( 1 - 5.35T + 29T^{2} \) |
| 31 | \( 1 - 5.61T + 31T^{2} \) |
| 37 | \( 1 + 2.41T + 37T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 - 5.86T + 47T^{2} \) |
| 53 | \( 1 - 1.35T + 53T^{2} \) |
| 59 | \( 1 + 4.31T + 59T^{2} \) |
| 61 | \( 1 - 4.57T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 8.71T + 79T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 8.70T + 97T^{2} \) |
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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
−11.74613948510870517130344674968, −10.17874980711735869748771492539, −9.625176995609073724697709839748, −8.698246988826019480647950599197, −7.84693273436023677328222707398, −6.39361284435998727634857309939, −5.58299392503635087811810900450, −4.32800161664540720907979597467, −3.41956822902228090603119168527, −1.31163640917120324403914216644,
1.31163640917120324403914216644, 3.41956822902228090603119168527, 4.32800161664540720907979597467, 5.58299392503635087811810900450, 6.39361284435998727634857309939, 7.84693273436023677328222707398, 8.698246988826019480647950599197, 9.625176995609073724697709839748, 10.17874980711735869748771492539, 11.74613948510870517130344674968
