| L(s) = 1 | + 2-s − 2.47·3-s + 4-s + 1.97·5-s − 2.47·6-s − 2.02·7-s + 8-s + 3.12·9-s + 1.97·10-s + 0.372·11-s − 2.47·12-s + 5.40·13-s − 2.02·14-s − 4.90·15-s + 16-s + 4.04·17-s + 3.12·18-s + 19-s + 1.97·20-s + 5.00·21-s + 0.372·22-s − 7.48·23-s − 2.47·24-s − 1.08·25-s + 5.40·26-s − 0.321·27-s − 2.02·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.42·3-s + 0.5·4-s + 0.885·5-s − 1.01·6-s − 0.764·7-s + 0.353·8-s + 1.04·9-s + 0.625·10-s + 0.112·11-s − 0.714·12-s + 1.49·13-s − 0.540·14-s − 1.26·15-s + 0.250·16-s + 0.981·17-s + 0.737·18-s + 0.229·19-s + 0.442·20-s + 1.09·21-s + 0.0793·22-s − 1.55·23-s − 0.505·24-s − 0.216·25-s + 1.05·26-s − 0.0618·27-s − 0.382·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.340380567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.340380567\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 - 1.97T + 5T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 11 | \( 1 - 0.372T + 11T^{2} \) |
| 13 | \( 1 - 5.40T + 13T^{2} \) |
| 17 | \( 1 - 4.04T + 17T^{2} \) |
| 23 | \( 1 + 7.48T + 23T^{2} \) |
| 29 | \( 1 + 0.453T + 29T^{2} \) |
| 31 | \( 1 - 4.29T + 31T^{2} \) |
| 37 | \( 1 - 0.878T + 37T^{2} \) |
| 41 | \( 1 - 7.17T + 41T^{2} \) |
| 43 | \( 1 + 0.577T + 43T^{2} \) |
| 47 | \( 1 - 6.26T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 - 1.83T + 59T^{2} \) |
| 61 | \( 1 - 6.58T + 61T^{2} \) |
| 67 | \( 1 + 5.10T + 67T^{2} \) |
| 71 | \( 1 - 2.48T + 71T^{2} \) |
| 73 | \( 1 - 3.20T + 73T^{2} \) |
| 79 | \( 1 + 3.49T + 79T^{2} \) |
| 83 | \( 1 + 8.10T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 0.679T + 97T^{2} \) |
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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
−7.59869904475750523224599200069, −6.65469882846229405861415727713, −6.19919252599977732825318580510, −5.76425684918637746683795177180, −5.42824275257381186826264768176, −4.31215804830235880339523359385, −3.72928013203618080254140222440, −2.76364991059590490847773314761, −1.65250256739172140712464084301, −0.76371986391137245416385228742,
0.76371986391137245416385228742, 1.65250256739172140712464084301, 2.76364991059590490847773314761, 3.72928013203618080254140222440, 4.31215804830235880339523359385, 5.42824275257381186826264768176, 5.76425684918637746683795177180, 6.19919252599977732825318580510, 6.65469882846229405861415727713, 7.59869904475750523224599200069
