L(s) = 1 | − 0.950·2-s + 3-s − 1.09·4-s − 0.0164·5-s − 0.950·6-s + 3.21·7-s + 2.94·8-s + 9-s + 0.0156·10-s + 1.39·11-s − 1.09·12-s + 13-s − 3.05·14-s − 0.0164·15-s − 0.605·16-s − 0.435·17-s − 0.950·18-s − 5.48·19-s + 0.0180·20-s + 3.21·21-s − 1.32·22-s − 3.31·23-s + 2.94·24-s − 4.99·25-s − 0.950·26-s + 27-s − 3.52·28-s + ⋯ |
L(s) = 1 | − 0.672·2-s + 0.577·3-s − 0.548·4-s − 0.00734·5-s − 0.388·6-s + 1.21·7-s + 1.04·8-s + 0.333·9-s + 0.00493·10-s + 0.420·11-s − 0.316·12-s + 0.277·13-s − 0.816·14-s − 0.00424·15-s − 0.151·16-s − 0.105·17-s − 0.224·18-s − 1.25·19-s + 0.00402·20-s + 0.701·21-s − 0.282·22-s − 0.691·23-s + 0.600·24-s − 0.999·25-s − 0.186·26-s + 0.192·27-s − 0.666·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 + 0.950T + 2T^{2} \) |
| 5 | \( 1 + 0.0164T + 5T^{2} \) |
| 7 | \( 1 - 3.21T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 17 | \( 1 + 0.435T + 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 + 3.31T + 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 + 2.13T + 31T^{2} \) |
| 37 | \( 1 + 9.14T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 0.975T + 43T^{2} \) |
| 47 | \( 1 - 5.71T + 47T^{2} \) |
| 53 | \( 1 - 1.19T + 53T^{2} \) |
| 59 | \( 1 - 4.72T + 59T^{2} \) |
| 61 | \( 1 + 6.39T + 61T^{2} \) |
| 67 | \( 1 + 2.71T + 67T^{2} \) |
| 71 | \( 1 + 8.17T + 71T^{2} \) |
| 73 | \( 1 + 0.674T + 73T^{2} \) |
| 79 | \( 1 + 6.74T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 4.39T + 89T^{2} \) |
| 97 | \( 1 - 6.62T + 97T^{2} \) |
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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
−8.181644883223109907416489315489, −7.68205484601160394049983695750, −6.90800339222623753017256366575, −5.79064906773975968322155857142, −4.97155303449331191089069200729, −4.14842827874491793667186631705, −3.63806576143301355474054689812, −1.97928225636264136161069573626, −1.61238877351282396033949669031, 0,
1.61238877351282396033949669031, 1.97928225636264136161069573626, 3.63806576143301355474054689812, 4.14842827874491793667186631705, 4.97155303449331191089069200729, 5.79064906773975968322155857142, 6.90800339222623753017256366575, 7.68205484601160394049983695750, 8.181644883223109907416489315489