| L(s) = 1 | − 2-s − 2.83·3-s + 4-s + 0.330·5-s + 2.83·6-s − 7-s − 8-s + 5.05·9-s − 0.330·10-s + 3.07·11-s − 2.83·12-s + 2.69·13-s + 14-s − 0.936·15-s + 16-s − 6.41·17-s − 5.05·18-s + 3.15·19-s + 0.330·20-s + 2.83·21-s − 3.07·22-s − 6.50·23-s + 2.83·24-s − 4.89·25-s − 2.69·26-s − 5.84·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.63·3-s + 0.5·4-s + 0.147·5-s + 1.15·6-s − 0.377·7-s − 0.353·8-s + 1.68·9-s − 0.104·10-s + 0.928·11-s − 0.819·12-s + 0.746·13-s + 0.267·14-s − 0.241·15-s + 0.250·16-s − 1.55·17-s − 1.19·18-s + 0.723·19-s + 0.0737·20-s + 0.619·21-s − 0.656·22-s − 1.35·23-s + 0.579·24-s − 0.978·25-s − 0.528·26-s − 1.12·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6325082220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6325082220\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 2.83T + 3T^{2} \) |
| 5 | \( 1 - 0.330T + 5T^{2} \) |
| 11 | \( 1 - 3.07T + 11T^{2} \) |
| 13 | \( 1 - 2.69T + 13T^{2} \) |
| 17 | \( 1 + 6.41T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 + 6.50T + 23T^{2} \) |
| 29 | \( 1 - 3.58T + 29T^{2} \) |
| 31 | \( 1 + 0.453T + 31T^{2} \) |
| 37 | \( 1 - 0.0743T + 37T^{2} \) |
| 41 | \( 1 - 9.21T + 41T^{2} \) |
| 43 | \( 1 - 7.47T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 3.91T + 53T^{2} \) |
| 59 | \( 1 - 7.17T + 59T^{2} \) |
| 61 | \( 1 + 7.56T + 61T^{2} \) |
| 67 | \( 1 - 1.60T + 67T^{2} \) |
| 71 | \( 1 + 5.69T + 71T^{2} \) |
| 73 | \( 1 + 6.01T + 73T^{2} \) |
| 79 | \( 1 + 5.72T + 79T^{2} \) |
| 83 | \( 1 - 9.56T + 83T^{2} \) |
| 89 | \( 1 + 8.49T + 89T^{2} \) |
| 97 | \( 1 - 1.71T + 97T^{2} \) |
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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
−8.015050801805333997999219662615, −7.14565036408826449910405058946, −6.60175405600989564573684461691, −5.93918567888322093237869955226, −5.67614720590483500766038016041, −4.34414097941987469208730280141, −3.97115994220480043279460330172, −2.50023319411368462537445397048, −1.44252676124240884623035271505, −0.54185854538306070155732138915,
0.54185854538306070155732138915, 1.44252676124240884623035271505, 2.50023319411368462537445397048, 3.97115994220480043279460330172, 4.34414097941987469208730280141, 5.67614720590483500766038016041, 5.93918567888322093237869955226, 6.60175405600989564573684461691, 7.14565036408826449910405058946, 8.015050801805333997999219662615
