| L(s) = 1 | − 2.55·2-s − 3-s + 4.54·4-s + 3.55·5-s + 2.55·6-s + 7-s − 6.51·8-s + 9-s − 9.08·10-s − 1.59·11-s − 4.54·12-s + 3.71·13-s − 2.55·14-s − 3.55·15-s + 7.57·16-s + 2.71·17-s − 2.55·18-s + 1.03·19-s + 16.1·20-s − 21-s + 4.09·22-s + 3.04·23-s + 6.51·24-s + 7.61·25-s − 9.51·26-s − 27-s + 4.54·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s − 0.577·3-s + 2.27·4-s + 1.58·5-s + 1.04·6-s + 0.377·7-s − 2.30·8-s + 0.333·9-s − 2.87·10-s − 0.482·11-s − 1.31·12-s + 1.03·13-s − 0.683·14-s − 0.917·15-s + 1.89·16-s + 0.658·17-s − 0.603·18-s + 0.238·19-s + 3.61·20-s − 0.218·21-s + 0.872·22-s + 0.635·23-s + 1.32·24-s + 1.52·25-s − 1.86·26-s − 0.192·27-s + 0.859·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.358824556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358824556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 5 | \( 1 - 3.55T + 5T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 - 3.71T + 13T^{2} \) |
| 17 | \( 1 - 2.71T + 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 - 3.04T + 23T^{2} \) |
| 29 | \( 1 - 7.73T + 29T^{2} \) |
| 31 | \( 1 - 8.27T + 31T^{2} \) |
| 37 | \( 1 - 6.82T + 37T^{2} \) |
| 41 | \( 1 - 0.500T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 6.65T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 7.73T + 59T^{2} \) |
| 61 | \( 1 - 6.96T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 - 1.90T + 73T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 + 16.1T + 83T^{2} \) |
| 89 | \( 1 + 3.81T + 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−8.084384568642451750881354163315, −7.18260686776389287078496082031, −6.54097280818629891343340791733, −5.96854223257899156377932738611, −5.46317482426591443543462576766, −4.45575460019266841675155464066, −2.91032713890889790994896807231, −2.36153484029967652959265652255, −1.24195868732942777491479797244, −0.968652393610117246833550909454,
0.968652393610117246833550909454, 1.24195868732942777491479797244, 2.36153484029967652959265652255, 2.91032713890889790994896807231, 4.45575460019266841675155464066, 5.46317482426591443543462576766, 5.96854223257899156377932738611, 6.54097280818629891343340791733, 7.18260686776389287078496082031, 8.084384568642451750881354163315
