| L(s) = 1 | − 0.296·2-s + 2.36·3-s − 1.91·4-s + 5-s − 0.703·6-s + 7-s + 1.16·8-s + 2.61·9-s − 0.296·10-s + 1.45·11-s − 4.53·12-s + 1.29·13-s − 0.296·14-s + 2.36·15-s + 3.47·16-s + 17-s − 0.776·18-s − 1.45·19-s − 1.91·20-s + 2.36·21-s − 0.432·22-s + 2.54·23-s + 2.75·24-s + 25-s − 0.384·26-s − 0.911·27-s − 1.91·28-s + ⋯ |
| L(s) = 1 | − 0.209·2-s + 1.36·3-s − 0.955·4-s + 0.447·5-s − 0.287·6-s + 0.377·7-s + 0.410·8-s + 0.871·9-s − 0.0938·10-s + 0.439·11-s − 1.30·12-s + 0.359·13-s − 0.0793·14-s + 0.611·15-s + 0.869·16-s + 0.242·17-s − 0.182·18-s − 0.334·19-s − 0.427·20-s + 0.517·21-s − 0.0922·22-s + 0.530·23-s + 0.561·24-s + 0.200·25-s − 0.0754·26-s − 0.175·27-s − 0.361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.955090113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.955090113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 0.296T + 2T^{2} \) |
| 3 | \( 1 - 2.36T + 3T^{2} \) |
| 11 | \( 1 - 1.45T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 19 | \( 1 + 1.45T + 19T^{2} \) |
| 23 | \( 1 - 2.54T + 23T^{2} \) |
| 29 | \( 1 + 0.212T + 29T^{2} \) |
| 31 | \( 1 - 7.81T + 31T^{2} \) |
| 37 | \( 1 - 1.03T + 37T^{2} \) |
| 41 | \( 1 - 3.89T + 41T^{2} \) |
| 43 | \( 1 + 0.703T + 43T^{2} \) |
| 47 | \( 1 + 8.46T + 47T^{2} \) |
| 53 | \( 1 + 4.62T + 53T^{2} \) |
| 59 | \( 1 - 8.24T + 59T^{2} \) |
| 61 | \( 1 + 4.31T + 61T^{2} \) |
| 67 | \( 1 + 5.97T + 67T^{2} \) |
| 71 | \( 1 + 9.87T + 71T^{2} \) |
| 73 | \( 1 + 1.07T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 8.95T + 89T^{2} \) |
| 97 | \( 1 + 8.45T + 97T^{2} \) |
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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
−10.28631210934656552159843682209, −9.590472016699255595301855654198, −8.838395276563277616554189643551, −8.337011326313763670140554947931, −7.49089283858640747646824465690, −6.16099319566124292303194308745, −4.88825134182498779911900655597, −3.93601914136006177934500819790, −2.85435758948977042621884932337, −1.42501856138959942649737817251,
1.42501856138959942649737817251, 2.85435758948977042621884932337, 3.93601914136006177934500819790, 4.88825134182498779911900655597, 6.16099319566124292303194308745, 7.49089283858640747646824465690, 8.337011326313763670140554947931, 8.838395276563277616554189643551, 9.590472016699255595301855654198, 10.28631210934656552159843682209
