| L(s) = 1 | − 2-s − 1.51·3-s + 4-s + 2.24·5-s + 1.51·6-s + 7-s − 8-s − 0.696·9-s − 2.24·10-s + 4.44·11-s − 1.51·12-s − 1.71·13-s − 14-s − 3.41·15-s + 16-s + 0.249·17-s + 0.696·18-s + 7.25·19-s + 2.24·20-s − 1.51·21-s − 4.44·22-s + 1.51·24-s + 0.0611·25-s + 1.71·26-s + 5.61·27-s + 28-s + 4.19·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.876·3-s + 0.5·4-s + 1.00·5-s + 0.619·6-s + 0.377·7-s − 0.353·8-s − 0.232·9-s − 0.711·10-s + 1.34·11-s − 0.438·12-s − 0.476·13-s − 0.267·14-s − 0.881·15-s + 0.250·16-s + 0.0605·17-s + 0.164·18-s + 1.66·19-s + 0.503·20-s − 0.331·21-s − 0.948·22-s + 0.309·24-s + 0.0122·25-s + 0.336·26-s + 1.07·27-s + 0.188·28-s + 0.779·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.541185862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.541185862\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 1.51T + 3T^{2} \) |
| 5 | \( 1 - 2.24T + 5T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 + 1.71T + 13T^{2} \) |
| 17 | \( 1 - 0.249T + 17T^{2} \) |
| 19 | \( 1 - 7.25T + 19T^{2} \) |
| 29 | \( 1 - 4.19T + 29T^{2} \) |
| 31 | \( 1 - 9.27T + 31T^{2} \) |
| 37 | \( 1 + 7.48T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 - 5.86T + 43T^{2} \) |
| 47 | \( 1 + 6.13T + 47T^{2} \) |
| 53 | \( 1 + 8.81T + 53T^{2} \) |
| 59 | \( 1 - 9.05T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 5.81T + 67T^{2} \) |
| 71 | \( 1 - 12.8T + 71T^{2} \) |
| 73 | \( 1 + 0.182T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 8.35T + 83T^{2} \) |
| 89 | \( 1 - 4.85T + 89T^{2} \) |
| 97 | \( 1 - 7.76T + 97T^{2} \) |
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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
−7.959907155257606510864558843735, −7.02056506481484628419763619481, −6.53692700771236939309451066366, −5.88656676344258423110990288815, −5.28910271159041393192005745339, −4.58733103726191877535396676705, −3.38195877159965462114251309808, −2.49443631604722549845412402310, −1.47279773319997229184653979207, −0.801016731639091983875451887299,
0.801016731639091983875451887299, 1.47279773319997229184653979207, 2.49443631604722549845412402310, 3.38195877159965462114251309808, 4.58733103726191877535396676705, 5.28910271159041393192005745339, 5.88656676344258423110990288815, 6.53692700771236939309451066366, 7.02056506481484628419763619481, 7.959907155257606510864558843735
