| L(s) = 1 | − 2.26·2-s + 2.87·3-s + 3.12·4-s + 0.727·5-s − 6.50·6-s + 0.231·7-s − 2.53·8-s + 5.26·9-s − 1.64·10-s − 3.33·11-s + 8.97·12-s + 13-s − 0.522·14-s + 2.09·15-s − 0.496·16-s − 3.22·17-s − 11.9·18-s + 6.87·19-s + 2.27·20-s + 0.664·21-s + 7.55·22-s + 5.68·23-s − 7.30·24-s − 4.47·25-s − 2.26·26-s + 6.51·27-s + 0.721·28-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.65·3-s + 1.56·4-s + 0.325·5-s − 2.65·6-s + 0.0873·7-s − 0.897·8-s + 1.75·9-s − 0.520·10-s − 1.00·11-s + 2.59·12-s + 0.277·13-s − 0.139·14-s + 0.539·15-s − 0.124·16-s − 0.781·17-s − 2.80·18-s + 1.57·19-s + 0.507·20-s + 0.144·21-s + 1.61·22-s + 1.18·23-s − 1.49·24-s − 0.894·25-s − 0.443·26-s + 1.25·27-s + 0.136·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 617 | \( 1 - T \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 - 0.727T + 5T^{2} \) |
| 7 | \( 1 - 0.231T + 7T^{2} \) |
| 11 | \( 1 + 3.33T + 11T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 - 6.87T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 + 7.88T + 31T^{2} \) |
| 37 | \( 1 + 0.678T + 37T^{2} \) |
| 41 | \( 1 + 1.78T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 6.89T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 - 3.71T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 2.94T + 67T^{2} \) |
| 71 | \( 1 - 4.67T + 71T^{2} \) |
| 73 | \( 1 + 9.34T + 73T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 1.49T + 83T^{2} \) |
| 89 | \( 1 + 7.30T + 89T^{2} \) |
| 97 | \( 1 - 5.41T + 97T^{2} \) |
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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
−7.69965058623920264919388063295, −7.32777278341188431780817628606, −6.57776029531932983160128928459, −5.43766049085585367125140633803, −4.60562101218049918184246312767, −3.33932179887435947141901338450, −2.97125881045898954059853328813, −1.88701950311568480716766426046, −1.56218323977260975227868127021, 0,
1.56218323977260975227868127021, 1.88701950311568480716766426046, 2.97125881045898954059853328813, 3.33932179887435947141901338450, 4.60562101218049918184246312767, 5.43766049085585367125140633803, 6.57776029531932983160128928459, 7.32777278341188431780817628606, 7.69965058623920264919388063295
