L(s) = 1 | + 2-s − 2.56·3-s + 4-s − 2.25·5-s − 2.56·6-s + 1.14·7-s + 8-s + 3.57·9-s − 2.25·10-s − 6.03·11-s − 2.56·12-s + 3.54·13-s + 1.14·14-s + 5.77·15-s + 16-s + 2.13·17-s + 3.57·18-s − 19-s − 2.25·20-s − 2.92·21-s − 6.03·22-s + 1.93·23-s − 2.56·24-s + 0.0736·25-s + 3.54·26-s − 1.48·27-s + 1.14·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.48·3-s + 0.5·4-s − 1.00·5-s − 1.04·6-s + 0.431·7-s + 0.353·8-s + 1.19·9-s − 0.712·10-s − 1.82·11-s − 0.740·12-s + 0.982·13-s + 0.304·14-s + 1.49·15-s + 0.250·16-s + 0.516·17-s + 0.843·18-s − 0.229·19-s − 0.503·20-s − 0.638·21-s − 1.28·22-s + 0.404·23-s − 0.523·24-s + 0.0147·25-s + 0.695·26-s − 0.285·27-s + 0.215·28-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 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 + 6.03T + 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 23 | \( 1 - 1.93T + 23T^{2} \) |
| 29 | \( 1 + 1.89T + 29T^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + 5.86T + 37T^{2} \) |
| 41 | \( 1 - 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.80T + 43T^{2} \) |
| 47 | \( 1 + 4.75T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 - 8.31T + 59T^{2} \) |
| 61 | \( 1 + 11.2T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 4.06T + 79T^{2} \) |
| 83 | \( 1 - 3.29T + 83T^{2} \) |
| 89 | \( 1 - 1.97T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.39943245528531107041539289393, −6.68709568037087016190273377031, −5.79649368205135987847468679888, −5.42619678613602586132849909357, −4.83561456027188221989840136288, −4.07975435795834111167207161174, −3.35887324730145002974379603617, −2.29922366029677593453289911822, −1.02748995653527698226411735156, 0,
1.02748995653527698226411735156, 2.29922366029677593453289911822, 3.35887324730145002974379603617, 4.07975435795834111167207161174, 4.83561456027188221989840136288, 5.42619678613602586132849909357, 5.79649368205135987847468679888, 6.68709568037087016190273377031, 7.39943245528531107041539289393