| L(s) = 1 | − 2.07·2-s + 2.30·4-s + 0.0743·5-s + 1.39·7-s − 0.628·8-s − 0.154·10-s − 6.37·11-s − 2.33·13-s − 2.90·14-s − 3.30·16-s + 0.377·17-s + 7.75·19-s + 0.171·20-s + 13.2·22-s − 5.47·23-s − 4.99·25-s + 4.83·26-s + 3.22·28-s + 8.56·29-s − 3.25·31-s + 8.10·32-s − 0.782·34-s + 0.104·35-s − 7.65·37-s − 16.0·38-s − 0.0466·40-s − 9.41·41-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.15·4-s + 0.0332·5-s + 0.528·7-s − 0.222·8-s − 0.0487·10-s − 1.92·11-s − 0.646·13-s − 0.775·14-s − 0.825·16-s + 0.0914·17-s + 1.77·19-s + 0.0382·20-s + 2.82·22-s − 1.14·23-s − 0.998·25-s + 0.948·26-s + 0.609·28-s + 1.59·29-s − 0.584·31-s + 1.43·32-s − 0.134·34-s + 0.0175·35-s − 1.25·37-s − 2.60·38-s − 0.00737·40-s − 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 + 2.07T + 2T^{2} \) |
| 5 | \( 1 - 0.0743T + 5T^{2} \) |
| 7 | \( 1 - 1.39T + 7T^{2} \) |
| 11 | \( 1 + 6.37T + 11T^{2} \) |
| 13 | \( 1 + 2.33T + 13T^{2} \) |
| 17 | \( 1 - 0.377T + 17T^{2} \) |
| 19 | \( 1 - 7.75T + 19T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 - 8.56T + 29T^{2} \) |
| 31 | \( 1 + 3.25T + 31T^{2} \) |
| 37 | \( 1 + 7.65T + 37T^{2} \) |
| 41 | \( 1 + 9.41T + 41T^{2} \) |
| 43 | \( 1 + 0.589T + 43T^{2} \) |
| 47 | \( 1 - 7.65T + 47T^{2} \) |
| 53 | \( 1 + 3.19T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 8.47T + 67T^{2} \) |
| 73 | \( 1 - 9.40T + 73T^{2} \) |
| 79 | \( 1 + 9.88T + 79T^{2} \) |
| 83 | \( 1 + 1.08T + 83T^{2} \) |
| 89 | \( 1 + 5.17T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−10.16148377423652139391419176284, −9.370036587801871350385541032613, −8.206426522125203781471927125076, −7.83858284051030987119081705337, −7.08638884338583378538172436977, −5.58901124701804123965224585002, −4.75195734550767851345065043628, −2.93585400480560733819336149381, −1.73619587757259194725251892724, 0,
1.73619587757259194725251892724, 2.93585400480560733819336149381, 4.75195734550767851345065043628, 5.58901124701804123965224585002, 7.08638884338583378538172436977, 7.83858284051030987119081705337, 8.206426522125203781471927125076, 9.370036587801871350385541032613, 10.16148377423652139391419176284
