| L(s) = 1 | + 2-s + 1.83·3-s + 4-s − 1.83·5-s + 1.83·6-s + 3.64·7-s + 8-s + 0.372·9-s − 1.83·10-s + 1.73·11-s + 1.83·12-s + 0.295·13-s + 3.64·14-s − 3.36·15-s + 16-s − 0.742·17-s + 0.372·18-s + 2.21·19-s − 1.83·20-s + 6.70·21-s + 1.73·22-s + 23-s + 1.83·24-s − 1.63·25-s + 0.295·26-s − 4.82·27-s + 3.64·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.06·3-s + 0.5·4-s − 0.819·5-s + 0.749·6-s + 1.37·7-s + 0.353·8-s + 0.124·9-s − 0.579·10-s + 0.524·11-s + 0.530·12-s + 0.0819·13-s + 0.975·14-s − 0.869·15-s + 0.250·16-s − 0.179·17-s + 0.0878·18-s + 0.508·19-s − 0.409·20-s + 1.46·21-s + 0.370·22-s + 0.208·23-s + 0.374·24-s − 0.327·25-s + 0.0579·26-s − 0.928·27-s + 0.689·28-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
| \(L(1)\) |
\(\approx\) |
\(5.043662272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.043662272\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.83T + 3T^{2} \) |
| 5 | \( 1 + 1.83T + 5T^{2} \) |
| 7 | \( 1 - 3.64T + 7T^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 - 0.295T + 13T^{2} \) |
| 17 | \( 1 + 0.742T + 17T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 29 | \( 1 - 8.68T + 29T^{2} \) |
| 31 | \( 1 + 2.15T + 31T^{2} \) |
| 37 | \( 1 + 4.71T + 37T^{2} \) |
| 41 | \( 1 - 0.183T + 41T^{2} \) |
| 43 | \( 1 - 11.5T + 43T^{2} \) |
| 47 | \( 1 - 3.49T + 47T^{2} \) |
| 53 | \( 1 - 5.08T + 53T^{2} \) |
| 59 | \( 1 - 6.03T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 1.91T + 67T^{2} \) |
| 71 | \( 1 + 3.56T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 6.97T + 79T^{2} \) |
| 83 | \( 1 + 8.14T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.167371293877378507691596989639, −7.43153139573183907341753004412, −6.88296533545783703908633733567, −5.74976709361842963812419206837, −5.07603814616689306163853901177, −4.17348519042411353237529753813, −3.83098477583432165895041438360, −2.84623023866184030707496849478, −2.12495704104482818354010947531, −1.07228383095053215330368763765,
1.07228383095053215330368763765, 2.12495704104482818354010947531, 2.84623023866184030707496849478, 3.83098477583432165895041438360, 4.17348519042411353237529753813, 5.07603814616689306163853901177, 5.74976709361842963812419206837, 6.88296533545783703908633733567, 7.43153139573183907341753004412, 8.167371293877378507691596989639
