| L(s) = 1 | − 2.14·2-s + 2.61·4-s − 2.14·5-s + 4.23·7-s − 1.32·8-s + 4.61·10-s − 0.236·13-s − 9.10·14-s − 2.38·16-s + 6.13·17-s + 2.38·19-s − 5.62·20-s + 3.98·23-s − 0.381·25-s + 0.507·26-s + 11.0·28-s − 6.95·29-s − 4.61·31-s + 7.77·32-s − 13.1·34-s − 9.10·35-s + 1.76·37-s − 5.11·38-s + 2.85·40-s + 2.96·41-s − 2.70·43-s − 8.56·46-s + ⋯ |
| L(s) = 1 | − 1.51·2-s + 1.30·4-s − 0.961·5-s + 1.60·7-s − 0.469·8-s + 1.46·10-s − 0.0654·13-s − 2.43·14-s − 0.595·16-s + 1.48·17-s + 0.546·19-s − 1.25·20-s + 0.830·23-s − 0.0763·25-s + 0.0994·26-s + 2.09·28-s − 1.29·29-s − 0.829·31-s + 1.37·32-s − 2.26·34-s − 1.53·35-s + 0.289·37-s − 0.830·38-s + 0.451·40-s + 0.463·41-s − 0.412·43-s − 1.26·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7682954276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7682954276\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 5 | \( 1 + 2.14T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 - 6.13T + 17T^{2} \) |
| 19 | \( 1 - 2.38T + 19T^{2} \) |
| 23 | \( 1 - 3.98T + 23T^{2} \) |
| 29 | \( 1 + 6.95T + 29T^{2} \) |
| 31 | \( 1 + 4.61T + 31T^{2} \) |
| 37 | \( 1 - 1.76T + 37T^{2} \) |
| 41 | \( 1 - 2.96T + 41T^{2} \) |
| 43 | \( 1 + 2.70T + 43T^{2} \) |
| 47 | \( 1 - 3.47T + 47T^{2} \) |
| 53 | \( 1 + 1.83T + 53T^{2} \) |
| 59 | \( 1 + 7.46T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 + 2.85T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 2.47T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 + 8.28T + 83T^{2} \) |
| 89 | \( 1 - 8.90T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−9.731427803056437242455279869836, −8.950503417063803954333526026097, −8.118132124693307734620253588527, −7.65725849357135015892261130120, −7.20814020140834849724457132232, −5.61048574523117506071229427643, −4.68507288050757197273699663287, −3.51844548917373724832957693963, −1.93803245836740233723487422194, −0.884872128432394247420050518438,
0.884872128432394247420050518438, 1.93803245836740233723487422194, 3.51844548917373724832957693963, 4.68507288050757197273699663287, 5.61048574523117506071229427643, 7.20814020140834849724457132232, 7.65725849357135015892261130120, 8.118132124693307734620253588527, 8.950503417063803954333526026097, 9.731427803056437242455279869836
