| L(s) = 1 | + 2-s + 1.36·3-s + 4-s + 4.43·5-s + 1.36·6-s + 7-s + 8-s − 1.12·9-s + 4.43·10-s + 3.73·11-s + 1.36·12-s + 1.16·13-s + 14-s + 6.07·15-s + 16-s + 1.48·17-s − 1.12·18-s − 6.21·19-s + 4.43·20-s + 1.36·21-s + 3.73·22-s − 0.805·23-s + 1.36·24-s + 14.7·25-s + 1.16·26-s − 5.64·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.790·3-s + 0.5·4-s + 1.98·5-s + 0.558·6-s + 0.377·7-s + 0.353·8-s − 0.375·9-s + 1.40·10-s + 1.12·11-s + 0.395·12-s + 0.323·13-s + 0.267·14-s + 1.56·15-s + 0.250·16-s + 0.359·17-s − 0.265·18-s − 1.42·19-s + 0.992·20-s + 0.298·21-s + 0.795·22-s − 0.168·23-s + 0.279·24-s + 2.94·25-s + 0.228·26-s − 1.08·27-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
| \(L(1)\) |
\(\approx\) |
\(6.797924024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.797924024\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.36T + 3T^{2} \) |
| 5 | \( 1 - 4.43T + 5T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 - 1.16T + 13T^{2} \) |
| 17 | \( 1 - 1.48T + 17T^{2} \) |
| 19 | \( 1 + 6.21T + 19T^{2} \) |
| 23 | \( 1 + 0.805T + 23T^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 + 6.87T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 - 4.40T + 41T^{2} \) |
| 43 | \( 1 + 8.02T + 43T^{2} \) |
| 47 | \( 1 + 2.06T + 47T^{2} \) |
| 53 | \( 1 - 9.49T + 53T^{2} \) |
| 59 | \( 1 + 9.99T + 59T^{2} \) |
| 61 | \( 1 - 0.0735T + 61T^{2} \) |
| 67 | \( 1 - 2.94T + 67T^{2} \) |
| 71 | \( 1 - 8.69T + 71T^{2} \) |
| 73 | \( 1 - 3.10T + 73T^{2} \) |
| 79 | \( 1 + 3.57T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 7.42T + 89T^{2} \) |
| 97 | \( 1 + 7.15T + 97T^{2} \) |
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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
−8.251179680734137232571658562223, −7.17338110133741095117904357711, −6.40633094433766880035813272426, −5.93505131729019430287982858819, −5.36867345669980397484218653663, −4.39933394542056731450808320665, −3.60035192160839158707389402828, −2.64898607531789071693977365493, −2.03556604164169935646414686525, −1.37572346360352169365790361842,
1.37572346360352169365790361842, 2.03556604164169935646414686525, 2.64898607531789071693977365493, 3.60035192160839158707389402828, 4.39933394542056731450808320665, 5.36867345669980397484218653663, 5.93505131729019430287982858819, 6.40633094433766880035813272426, 7.17338110133741095117904357711, 8.251179680734137232571658562223
