| L(s) = 1 | + 0.400·2-s − 1.83·4-s + 1.54·5-s − 0.0740·7-s − 1.53·8-s + 0.617·10-s − 2.80·11-s + 13-s − 0.0296·14-s + 3.06·16-s + 3.02·17-s + 4.58·19-s − 2.83·20-s − 1.12·22-s + 6.24·23-s − 2.61·25-s + 0.400·26-s + 0.136·28-s + 4.78·29-s + 5.41·31-s + 4.29·32-s + 1.20·34-s − 0.114·35-s − 2.01·37-s + 1.83·38-s − 2.37·40-s + 0.0386·41-s + ⋯ |
| L(s) = 1 | + 0.282·2-s − 0.919·4-s + 0.690·5-s − 0.0279·7-s − 0.543·8-s + 0.195·10-s − 0.844·11-s + 0.277·13-s − 0.00792·14-s + 0.766·16-s + 0.733·17-s + 1.05·19-s − 0.635·20-s − 0.238·22-s + 1.30·23-s − 0.523·25-s + 0.0784·26-s + 0.0257·28-s + 0.888·29-s + 0.971·31-s + 0.760·32-s + 0.207·34-s − 0.0193·35-s − 0.330·37-s + 0.297·38-s − 0.374·40-s + 0.00603·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1053 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655286194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655286194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 0.400T + 2T^{2} \) |
| 5 | \( 1 - 1.54T + 5T^{2} \) |
| 7 | \( 1 + 0.0740T + 7T^{2} \) |
| 11 | \( 1 + 2.80T + 11T^{2} \) |
| 17 | \( 1 - 3.02T + 17T^{2} \) |
| 19 | \( 1 - 4.58T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 - 4.78T + 29T^{2} \) |
| 31 | \( 1 - 5.41T + 31T^{2} \) |
| 37 | \( 1 + 2.01T + 37T^{2} \) |
| 41 | \( 1 - 0.0386T + 41T^{2} \) |
| 43 | \( 1 + 7.96T + 43T^{2} \) |
| 47 | \( 1 - 13.6T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 - 9.70T + 61T^{2} \) |
| 67 | \( 1 - 8.13T + 67T^{2} \) |
| 71 | \( 1 + 2.37T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 5.33T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−9.948901151519398854294329851627, −9.141318948118313873745779804748, −8.350807876169437402010184212390, −7.49845050950849663728962136116, −6.32365054554868448758065502267, −5.38659493214312902727559192058, −4.93780767635993490352755158609, −3.63666197021720050586182319598, −2.70174527525861033376270586442, −1.00353790974562063929348967726,
1.00353790974562063929348967726, 2.70174527525861033376270586442, 3.63666197021720050586182319598, 4.93780767635993490352755158609, 5.38659493214312902727559192058, 6.32365054554868448758065502267, 7.49845050950849663728962136116, 8.350807876169437402010184212390, 9.141318948118313873745779804748, 9.948901151519398854294329851627
