| L(s) = 1 | + 2-s + 1.47·3-s + 4-s + 1.86·5-s + 1.47·6-s + 0.679·7-s + 8-s − 0.823·9-s + 1.86·10-s + 1.27·11-s + 1.47·12-s + 3.42·13-s + 0.679·14-s + 2.75·15-s + 16-s + 3.16·17-s − 0.823·18-s + 6.32·19-s + 1.86·20-s + 1.00·21-s + 1.27·22-s + 23-s + 1.47·24-s − 1.52·25-s + 3.42·26-s − 5.64·27-s + 0.679·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.851·3-s + 0.5·4-s + 0.833·5-s + 0.602·6-s + 0.256·7-s + 0.353·8-s − 0.274·9-s + 0.589·10-s + 0.384·11-s + 0.425·12-s + 0.948·13-s + 0.181·14-s + 0.710·15-s + 0.250·16-s + 0.767·17-s − 0.194·18-s + 1.45·19-s + 0.416·20-s + 0.218·21-s + 0.272·22-s + 0.208·23-s + 0.301·24-s − 0.304·25-s + 0.670·26-s − 1.08·27-s + 0.128·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.843987918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.843987918\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.47T + 3T^{2} \) |
| 5 | \( 1 - 1.86T + 5T^{2} \) |
| 7 | \( 1 - 0.679T + 7T^{2} \) |
| 11 | \( 1 - 1.27T + 11T^{2} \) |
| 13 | \( 1 - 3.42T + 13T^{2} \) |
| 17 | \( 1 - 3.16T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 29 | \( 1 + 3.93T + 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 - 7.96T + 37T^{2} \) |
| 41 | \( 1 - 6.86T + 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 + 2.42T + 47T^{2} \) |
| 53 | \( 1 - 11.1T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 + 0.612T + 61T^{2} \) |
| 67 | \( 1 + 3.21T + 67T^{2} \) |
| 71 | \( 1 + 12.5T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 2.76T + 79T^{2} \) |
| 83 | \( 1 - 8.24T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−7.81375689327451090697459400740, −7.60361228655401529933342040462, −6.48106415995489480301910893917, −5.73585158213791778992825574345, −5.44922754615838646486950259718, −4.30896004989614730181068736109, −3.48095714139877156771116383243, −2.97514912864778405869061440906, −1.98902709908006751477048226944, −1.22246938030877769725931942489,
1.22246938030877769725931942489, 1.98902709908006751477048226944, 2.97514912864778405869061440906, 3.48095714139877156771116383243, 4.30896004989614730181068736109, 5.44922754615838646486950259718, 5.73585158213791778992825574345, 6.48106415995489480301910893917, 7.60361228655401529933342040462, 7.81375689327451090697459400740
