| L(s) = 1 | − 2.57·2-s − 3-s + 4.62·4-s − 4.27·5-s + 2.57·6-s − 4.90·7-s − 6.76·8-s + 9-s + 11.0·10-s + 2.43·11-s − 4.62·12-s + 3.14·13-s + 12.6·14-s + 4.27·15-s + 8.15·16-s − 6.78·17-s − 2.57·18-s − 5.14·19-s − 19.7·20-s + 4.90·21-s − 6.26·22-s − 6.13·23-s + 6.76·24-s + 13.3·25-s − 8.08·26-s − 27-s − 22.6·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s − 0.577·3-s + 2.31·4-s − 1.91·5-s + 1.05·6-s − 1.85·7-s − 2.39·8-s + 0.333·9-s + 3.48·10-s + 0.733·11-s − 1.33·12-s + 0.871·13-s + 3.37·14-s + 1.10·15-s + 2.03·16-s − 1.64·17-s − 0.606·18-s − 1.17·19-s − 4.42·20-s + 1.07·21-s − 1.33·22-s − 1.27·23-s + 1.38·24-s + 2.66·25-s − 1.58·26-s − 0.192·27-s − 4.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2671 | \( 1 + T \) |
| good | 2 | \( 1 + 2.57T + 2T^{2} \) |
| 5 | \( 1 + 4.27T + 5T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 + 6.78T + 17T^{2} \) |
| 19 | \( 1 + 5.14T + 19T^{2} \) |
| 23 | \( 1 + 6.13T + 23T^{2} \) |
| 29 | \( 1 + 8.16T + 29T^{2} \) |
| 31 | \( 1 - 3.92T + 31T^{2} \) |
| 37 | \( 1 - 0.264T + 37T^{2} \) |
| 41 | \( 1 + 3.49T + 41T^{2} \) |
| 43 | \( 1 + 9.08T + 43T^{2} \) |
| 47 | \( 1 - 2.42T + 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 + 8.90T + 67T^{2} \) |
| 71 | \( 1 + 9.27T + 71T^{2} \) |
| 73 | \( 1 + 0.319T + 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 1.53T + 89T^{2} \) |
| 97 | \( 1 - 12.8T + 97T^{2} \) |
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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
−7.51999627613131392451923606380, −6.85137447272374555352902532504, −6.54821801953506948175264297547, −5.97975134541069692100587029404, −4.28052636718115726458973037692, −3.85875303757401873624512054102, −3.03509329260632054846700900844, −1.89120498542820665161821615472, −0.53890675910827310678075770255, 0,
0.53890675910827310678075770255, 1.89120498542820665161821615472, 3.03509329260632054846700900844, 3.85875303757401873624512054102, 4.28052636718115726458973037692, 5.97975134541069692100587029404, 6.54821801953506948175264297547, 6.85137447272374555352902532504, 7.51999627613131392451923606380
