| L(s) = 1 | + 2-s − 1.53·3-s + 4-s + 3.94·5-s − 1.53·6-s − 3.54·7-s + 8-s − 0.633·9-s + 3.94·10-s + 4.54·11-s − 1.53·12-s + 3.47·13-s − 3.54·14-s − 6.06·15-s + 16-s + 3.19·17-s − 0.633·18-s + 1.14·19-s + 3.94·20-s + 5.44·21-s + 4.54·22-s − 3.21·23-s − 1.53·24-s + 10.5·25-s + 3.47·26-s + 5.58·27-s − 3.54·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.888·3-s + 0.5·4-s + 1.76·5-s − 0.628·6-s − 1.33·7-s + 0.353·8-s − 0.211·9-s + 1.24·10-s + 1.36·11-s − 0.444·12-s + 0.962·13-s − 0.946·14-s − 1.56·15-s + 0.250·16-s + 0.775·17-s − 0.149·18-s + 0.263·19-s + 0.881·20-s + 1.18·21-s + 0.968·22-s − 0.669·23-s − 0.314·24-s + 2.11·25-s + 0.680·26-s + 1.07·27-s − 0.669·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.039658414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.039658414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 + 1.53T + 3T^{2} \) |
| 5 | \( 1 - 3.94T + 5T^{2} \) |
| 7 | \( 1 + 3.54T + 7T^{2} \) |
| 11 | \( 1 - 4.54T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 - 1.14T + 19T^{2} \) |
| 23 | \( 1 + 3.21T + 23T^{2} \) |
| 29 | \( 1 - 4.23T + 29T^{2} \) |
| 31 | \( 1 + 8.64T + 31T^{2} \) |
| 37 | \( 1 + 4.09T + 37T^{2} \) |
| 41 | \( 1 + 5.96T + 41T^{2} \) |
| 43 | \( 1 - 7.60T + 43T^{2} \) |
| 47 | \( 1 - 4.18T + 47T^{2} \) |
| 53 | \( 1 + 0.668T + 53T^{2} \) |
| 59 | \( 1 + 0.928T + 59T^{2} \) |
| 61 | \( 1 - 8.93T + 61T^{2} \) |
| 67 | \( 1 + 4.43T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 9.97T + 83T^{2} \) |
| 89 | \( 1 - 8.67T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.797580865349008125088737689087, −7.24068658054593401236823069508, −6.49135447480403296445585273064, −6.04900285412185439320637692083, −5.80862748300356474807483119298, −4.95723175635435968290139970024, −3.74128001949198851930062636571, −3.12020687558114004373067424447, −1.96227638465860069166618783977, −0.985199034009814455327452467428,
0.985199034009814455327452467428, 1.96227638465860069166618783977, 3.12020687558114004373067424447, 3.74128001949198851930062636571, 4.95723175635435968290139970024, 5.80862748300356474807483119298, 6.04900285412185439320637692083, 6.49135447480403296445585273064, 7.24068658054593401236823069508, 8.797580865349008125088737689087
