| L(s) = 1 | − 2.43·2-s − 0.327·3-s + 3.94·4-s − 2.07·5-s + 0.798·6-s − 0.184·7-s − 4.73·8-s − 2.89·9-s + 5.06·10-s + 6.19·11-s − 1.29·12-s + 5.93·13-s + 0.448·14-s + 0.679·15-s + 3.65·16-s − 1.22·17-s + 7.05·18-s + 6.79·19-s − 8.18·20-s + 0.0602·21-s − 15.0·22-s + 4.78·23-s + 1.54·24-s − 0.689·25-s − 14.4·26-s + 1.93·27-s − 0.725·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.189·3-s + 1.97·4-s − 0.928·5-s + 0.325·6-s − 0.0695·7-s − 1.67·8-s − 0.964·9-s + 1.60·10-s + 1.86·11-s − 0.372·12-s + 1.64·13-s + 0.119·14-s + 0.175·15-s + 0.912·16-s − 0.296·17-s + 1.66·18-s + 1.55·19-s − 1.82·20-s + 0.0131·21-s − 3.21·22-s + 0.998·23-s + 0.316·24-s − 0.137·25-s − 2.83·26-s + 0.371·27-s − 0.137·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.43T + 2T^{2} \) |
| 3 | \( 1 + 0.327T + 3T^{2} \) |
| 5 | \( 1 + 2.07T + 5T^{2} \) |
| 7 | \( 1 + 0.184T + 7T^{2} \) |
| 11 | \( 1 - 6.19T + 11T^{2} \) |
| 13 | \( 1 - 5.93T + 13T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 19 | \( 1 - 6.79T + 19T^{2} \) |
| 23 | \( 1 - 4.78T + 23T^{2} \) |
| 29 | \( 1 + 1.27T + 29T^{2} \) |
| 31 | \( 1 + 3.43T + 31T^{2} \) |
| 37 | \( 1 - 6.19T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4.48T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 0.000567T + 59T^{2} \) |
| 61 | \( 1 + 3.47T + 61T^{2} \) |
| 67 | \( 1 - 0.649T + 67T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 + 0.0342T + 89T^{2} \) |
| 97 | \( 1 + 4.90T + 97T^{2} \) |
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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
−7.75653812531627464583109640628, −6.82653569550628245319692733173, −6.53459468995851242373811386358, −5.73890070991785600373111223234, −4.62033866661036671741606100422, −3.44247049300367600373959747968, −3.25490925301354972465594932961, −1.62644941272174030077036617839, −1.10867354828650742235012657409, 0,
1.10867354828650742235012657409, 1.62644941272174030077036617839, 3.25490925301354972465594932961, 3.44247049300367600373959747968, 4.62033866661036671741606100422, 5.73890070991785600373111223234, 6.53459468995851242373811386358, 6.82653569550628245319692733173, 7.75653812531627464583109640628
