| L(s) = 1 | − 2.12·2-s + 0.347·3-s + 2.51·4-s − 4.40·5-s − 0.738·6-s − 0.0796·7-s − 1.09·8-s − 2.87·9-s + 9.35·10-s + 4.35·11-s + 0.874·12-s − 13-s + 0.169·14-s − 1.53·15-s − 2.70·16-s − 0.234·17-s + 6.11·18-s + 7.31·19-s − 11.0·20-s − 0.0276·21-s − 9.24·22-s − 0.812·23-s − 0.379·24-s + 14.3·25-s + 2.12·26-s − 2.04·27-s − 0.200·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 0.200·3-s + 1.25·4-s − 1.96·5-s − 0.301·6-s − 0.0300·7-s − 0.386·8-s − 0.959·9-s + 2.95·10-s + 1.31·11-s + 0.252·12-s − 0.277·13-s + 0.0452·14-s − 0.395·15-s − 0.676·16-s − 0.0567·17-s + 1.44·18-s + 1.67·19-s − 2.47·20-s − 0.00604·21-s − 1.97·22-s − 0.169·23-s − 0.0775·24-s + 2.87·25-s + 0.416·26-s − 0.393·27-s − 0.0378·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 3 | \( 1 - 0.347T + 3T^{2} \) |
| 5 | \( 1 + 4.40T + 5T^{2} \) |
| 7 | \( 1 + 0.0796T + 7T^{2} \) |
| 11 | \( 1 - 4.35T + 11T^{2} \) |
| 17 | \( 1 + 0.234T + 17T^{2} \) |
| 19 | \( 1 - 7.31T + 19T^{2} \) |
| 23 | \( 1 + 0.812T + 23T^{2} \) |
| 29 | \( 1 + 2.12T + 29T^{2} \) |
| 31 | \( 1 + 3.40T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 7.15T + 41T^{2} \) |
| 43 | \( 1 - 6.12T + 43T^{2} \) |
| 47 | \( 1 + 0.190T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 2.95T + 59T^{2} \) |
| 61 | \( 1 + 7.49T + 61T^{2} \) |
| 67 | \( 1 + 8.85T + 67T^{2} \) |
| 71 | \( 1 - 7.80T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 8.22T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 4.38T + 89T^{2} \) |
| 97 | \( 1 - 1.24T + 97T^{2} \) |
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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
−9.147516745898920734773199453201, −8.397987614629158480775206500415, −7.72951361912113743456155090544, −7.31428268683553632584041082923, −6.30768170723437484293244442130, −4.81371157899498555600382448881, −3.79351329858149934891437467136, −2.93911390841042841555845693394, −1.19380940500905709465774766325, 0,
1.19380940500905709465774766325, 2.93911390841042841555845693394, 3.79351329858149934891437467136, 4.81371157899498555600382448881, 6.30768170723437484293244442130, 7.31428268683553632584041082923, 7.72951361912113743456155090544, 8.397987614629158480775206500415, 9.147516745898920734773199453201
