| L(s) = 1 | + 2-s − 1.23·3-s + 4-s − 0.854·5-s − 1.23·6-s + 4.26·7-s + 8-s − 1.47·9-s − 0.854·10-s + 4.13·11-s − 1.23·12-s − 5.17·13-s + 4.26·14-s + 1.05·15-s + 16-s + 4.83·17-s − 1.47·18-s − 5.48·19-s − 0.854·20-s − 5.26·21-s + 4.13·22-s + 5.47·23-s − 1.23·24-s − 4.26·25-s − 5.17·26-s + 5.52·27-s + 4.26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.713·3-s + 0.5·4-s − 0.382·5-s − 0.504·6-s + 1.61·7-s + 0.353·8-s − 0.491·9-s − 0.270·10-s + 1.24·11-s − 0.356·12-s − 1.43·13-s + 1.13·14-s + 0.272·15-s + 0.250·16-s + 1.17·17-s − 0.347·18-s − 1.25·19-s − 0.191·20-s − 1.14·21-s + 0.881·22-s + 1.14·23-s − 0.252·24-s − 0.853·25-s − 1.01·26-s + 1.06·27-s + 0.805·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.754890777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.754890777\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3023 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 5 | \( 1 + 0.854T + 5T^{2} \) |
| 7 | \( 1 - 4.26T + 7T^{2} \) |
| 11 | \( 1 - 4.13T + 11T^{2} \) |
| 13 | \( 1 + 5.17T + 13T^{2} \) |
| 17 | \( 1 - 4.83T + 17T^{2} \) |
| 19 | \( 1 + 5.48T + 19T^{2} \) |
| 23 | \( 1 - 5.47T + 23T^{2} \) |
| 29 | \( 1 + 0.234T + 29T^{2} \) |
| 31 | \( 1 - 4.50T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 - 0.532T + 43T^{2} \) |
| 47 | \( 1 + 8.38T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 4.07T + 59T^{2} \) |
| 61 | \( 1 - 7.14T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 1.35T + 71T^{2} \) |
| 73 | \( 1 - 1.42T + 73T^{2} \) |
| 79 | \( 1 - 5.33T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 12.5T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.102529421817818486862034659898, −7.20289166004624760785920960285, −6.61463564999056580879847840197, −5.74643611219223511619483102899, −5.11192407388679323961657003412, −4.59528913263827322890961662264, −3.92447893976786289159171370294, −2.80829989154478669269361790239, −1.87510265201788319591429455614, −0.837043175054785497291644940581,
0.837043175054785497291644940581, 1.87510265201788319591429455614, 2.80829989154478669269361790239, 3.92447893976786289159171370294, 4.59528913263827322890961662264, 5.11192407388679323961657003412, 5.74643611219223511619483102899, 6.61463564999056580879847840197, 7.20289166004624760785920960285, 8.102529421817818486862034659898
