| L(s) = 1 | − 2-s + 1.86·3-s + 4-s − 2.09·5-s − 1.86·6-s + 2.42·7-s − 8-s + 0.487·9-s + 2.09·10-s + 2.99·11-s + 1.86·12-s − 5.37·13-s − 2.42·14-s − 3.91·15-s + 16-s + 0.964·17-s − 0.487·18-s − 19-s − 2.09·20-s + 4.53·21-s − 2.99·22-s + 3.99·23-s − 1.86·24-s − 0.615·25-s + 5.37·26-s − 4.69·27-s + 2.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.07·3-s + 0.5·4-s − 0.936·5-s − 0.762·6-s + 0.917·7-s − 0.353·8-s + 0.162·9-s + 0.662·10-s + 0.902·11-s + 0.539·12-s − 1.49·13-s − 0.648·14-s − 1.00·15-s + 0.250·16-s + 0.233·17-s − 0.114·18-s − 0.229·19-s − 0.468·20-s + 0.989·21-s − 0.638·22-s + 0.832·23-s − 0.381·24-s − 0.123·25-s + 1.05·26-s − 0.903·27-s + 0.458·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.86T + 3T^{2} \) |
| 5 | \( 1 + 2.09T + 5T^{2} \) |
| 7 | \( 1 - 2.42T + 7T^{2} \) |
| 11 | \( 1 - 2.99T + 11T^{2} \) |
| 13 | \( 1 + 5.37T + 13T^{2} \) |
| 17 | \( 1 - 0.964T + 17T^{2} \) |
| 23 | \( 1 - 3.99T + 23T^{2} \) |
| 29 | \( 1 + 0.375T + 29T^{2} \) |
| 31 | \( 1 + 4.16T + 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 0.401T + 41T^{2} \) |
| 43 | \( 1 - 1.78T + 43T^{2} \) |
| 47 | \( 1 + 4.63T + 47T^{2} \) |
| 53 | \( 1 + 8.02T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 8.96T + 71T^{2} \) |
| 73 | \( 1 - 3.29T + 73T^{2} \) |
| 79 | \( 1 + 15.5T + 79T^{2} \) |
| 83 | \( 1 - 0.126T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 16.1T + 97T^{2} \) |
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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
−7.56817710189475569719541305807, −7.30992090000583085282508675361, −6.35865162616602020956127879605, −5.29124927572909872104820483853, −4.50584991318164965225938417021, −3.78398256576307826038446149799, −2.96865620683968879357230114326, −2.22090416200052067532163426854, −1.33877795295000928203890386999, 0,
1.33877795295000928203890386999, 2.22090416200052067532163426854, 2.96865620683968879357230114326, 3.78398256576307826038446149799, 4.50584991318164965225938417021, 5.29124927572909872104820483853, 6.35865162616602020956127879605, 7.30992090000583085282508675361, 7.56817710189475569719541305807
