| L(s) = 1 | − 2-s − 3-s + 4-s + 1.24·5-s + 6-s + 1.54·7-s − 8-s + 9-s − 1.24·10-s + 1.68·11-s − 12-s + 3.78·13-s − 1.54·14-s − 1.24·15-s + 16-s − 2.39·17-s − 18-s − 2.71·19-s + 1.24·20-s − 1.54·21-s − 1.68·22-s − 2.91·23-s + 24-s − 3.44·25-s − 3.78·26-s − 27-s + 1.54·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.558·5-s + 0.408·6-s + 0.583·7-s − 0.353·8-s + 0.333·9-s − 0.394·10-s + 0.508·11-s − 0.288·12-s + 1.05·13-s − 0.412·14-s − 0.322·15-s + 0.250·16-s − 0.581·17-s − 0.235·18-s − 0.623·19-s + 0.279·20-s − 0.337·21-s − 0.359·22-s − 0.608·23-s + 0.204·24-s − 0.688·25-s − 0.743·26-s − 0.192·27-s + 0.291·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5766 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 31 | \( 1 \) |
| good | 5 | \( 1 - 1.24T + 5T^{2} \) |
| 7 | \( 1 - 1.54T + 7T^{2} \) |
| 11 | \( 1 - 1.68T + 11T^{2} \) |
| 13 | \( 1 - 3.78T + 13T^{2} \) |
| 17 | \( 1 + 2.39T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 23 | \( 1 + 2.91T + 23T^{2} \) |
| 29 | \( 1 - 5.24T + 29T^{2} \) |
| 37 | \( 1 + 6.63T + 37T^{2} \) |
| 41 | \( 1 + 2.10T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 5.09T + 47T^{2} \) |
| 53 | \( 1 + 8.16T + 53T^{2} \) |
| 59 | \( 1 + 3.76T + 59T^{2} \) |
| 61 | \( 1 + 15.3T + 61T^{2} \) |
| 67 | \( 1 - 0.962T + 67T^{2} \) |
| 71 | \( 1 + 2.03T + 71T^{2} \) |
| 73 | \( 1 + 0.762T + 73T^{2} \) |
| 79 | \( 1 + 9.20T + 79T^{2} \) |
| 83 | \( 1 + 8.52T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 3.30T + 97T^{2} \) |
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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
−7.88828323800221221607382646443, −6.95777284111678907691337954320, −6.30726486746094182587004250165, −5.90820482211981000109907659121, −4.90702808138139022884198694251, −4.16458092660312299176340983926, −3.15110598303531371677425996214, −1.86483660906014034550341192858, −1.43781364320982337439866542595, 0,
1.43781364320982337439866542595, 1.86483660906014034550341192858, 3.15110598303531371677425996214, 4.16458092660312299176340983926, 4.90702808138139022884198694251, 5.90820482211981000109907659121, 6.30726486746094182587004250165, 6.95777284111678907691337954320, 7.88828323800221221607382646443
