L(s) = 1 | − 2-s − 3.09·3-s + 4-s − 2.61·5-s + 3.09·6-s − 2.12·7-s − 8-s + 6.60·9-s + 2.61·10-s + 0.148·11-s − 3.09·12-s + 3.47·13-s + 2.12·14-s + 8.11·15-s + 16-s − 5.38·17-s − 6.60·18-s + 5.56·19-s − 2.61·20-s + 6.59·21-s − 0.148·22-s + 3.85·23-s + 3.09·24-s + 1.86·25-s − 3.47·26-s − 11.1·27-s − 2.12·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.78·3-s + 0.5·4-s − 1.17·5-s + 1.26·6-s − 0.804·7-s − 0.353·8-s + 2.20·9-s + 0.828·10-s + 0.0448·11-s − 0.894·12-s + 0.964·13-s + 0.568·14-s + 2.09·15-s + 0.250·16-s − 1.30·17-s − 1.55·18-s + 1.27·19-s − 0.585·20-s + 1.43·21-s − 0.0317·22-s + 0.803·23-s + 0.632·24-s + 0.372·25-s − 0.681·26-s − 2.15·27-s − 0.402·28-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
\(L(1)\) |
\(\approx\) |
\(0.2913633775\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2913633775\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.09T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 7 | \( 1 + 2.12T + 7T^{2} \) |
| 11 | \( 1 - 0.148T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 - 5.56T + 19T^{2} \) |
| 23 | \( 1 - 3.85T + 23T^{2} \) |
| 29 | \( 1 + 5.49T + 29T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 - 9.30T + 37T^{2} \) |
| 41 | \( 1 - 4.64T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 2.74T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 + 5.84T + 71T^{2} \) |
| 73 | \( 1 - 8.90T + 73T^{2} \) |
| 79 | \( 1 + 5.50T + 79T^{2} \) |
| 83 | \( 1 + 9.90T + 83T^{2} \) |
| 89 | \( 1 - 2.44T + 89T^{2} \) |
| 97 | \( 1 + 5.79T + 97T^{2} \) |
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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
−8.360466903852611994269797608653, −7.53263093548936373034867411752, −6.93325685495148128000513798470, −6.32975774258812654422671795043, −5.70955796596622182535378714687, −4.71183748773266645452983551019, −3.98436888095940649323703754979, −3.03983995452010152129450565690, −1.36398441867754809336152756794, −0.41265177425984746368356071682,
0.41265177425984746368356071682, 1.36398441867754809336152756794, 3.03983995452010152129450565690, 3.98436888095940649323703754979, 4.71183748773266645452983551019, 5.70955796596622182535378714687, 6.32975774258812654422671795043, 6.93325685495148128000513798470, 7.53263093548936373034867411752, 8.360466903852611994269797608653