| L(s) = 1 | − 0.650·2-s − 1.57·4-s − 3.22·5-s + 3.97·7-s + 2.32·8-s + 2.09·10-s + 0.733·11-s − 0.452·13-s − 2.58·14-s + 1.64·16-s + 4.35·17-s + 2.01·19-s + 5.08·20-s − 0.476·22-s − 6.39·23-s + 5.38·25-s + 0.294·26-s − 6.27·28-s − 8.70·29-s − 5.41·31-s − 5.71·32-s − 2.83·34-s − 12.8·35-s − 0.547·37-s − 1.31·38-s − 7.49·40-s − 2.33·41-s + ⋯ |
| L(s) = 1 | − 0.459·2-s − 0.788·4-s − 1.44·5-s + 1.50·7-s + 0.822·8-s + 0.662·10-s + 0.221·11-s − 0.125·13-s − 0.690·14-s + 0.410·16-s + 1.05·17-s + 0.462·19-s + 1.13·20-s − 0.101·22-s − 1.33·23-s + 1.07·25-s + 0.0577·26-s − 1.18·28-s − 1.61·29-s − 0.972·31-s − 1.01·32-s − 0.485·34-s − 2.16·35-s − 0.0899·37-s − 0.212·38-s − 1.18·40-s − 0.364·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9181329243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9181329243\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 0.650T + 2T^{2} \) |
| 5 | \( 1 + 3.22T + 5T^{2} \) |
| 7 | \( 1 - 3.97T + 7T^{2} \) |
| 11 | \( 1 - 0.733T + 11T^{2} \) |
| 13 | \( 1 + 0.452T + 13T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 19 | \( 1 - 2.01T + 19T^{2} \) |
| 23 | \( 1 + 6.39T + 23T^{2} \) |
| 29 | \( 1 + 8.70T + 29T^{2} \) |
| 31 | \( 1 + 5.41T + 31T^{2} \) |
| 37 | \( 1 + 0.547T + 37T^{2} \) |
| 41 | \( 1 + 2.33T + 41T^{2} \) |
| 43 | \( 1 - 12.8T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 - 8.66T + 53T^{2} \) |
| 59 | \( 1 - 7.36T + 59T^{2} \) |
| 61 | \( 1 + 1.63T + 61T^{2} \) |
| 67 | \( 1 + 5.86T + 67T^{2} \) |
| 71 | \( 1 - 2.81T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 6.09T + 83T^{2} \) |
| 89 | \( 1 - 13.6T + 89T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
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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
−8.948855294072806600329807155442, −8.113773474535566665189760328556, −7.76061660707921004267918037138, −7.30723463231990436020555027108, −5.65201674854216042035134282380, −5.01346384154852895845410932385, −4.04329609354645315308990758107, −3.68805301534914151154218539274, −1.88326324805813234694592133263, −0.70143497148084560055447732391,
0.70143497148084560055447732391, 1.88326324805813234694592133263, 3.68805301534914151154218539274, 4.04329609354645315308990758107, 5.01346384154852895845410932385, 5.65201674854216042035134282380, 7.30723463231990436020555027108, 7.76061660707921004267918037138, 8.113773474535566665189760328556, 8.948855294072806600329807155442
