| L(s) = 1 | + 2.25·2-s − 3-s + 3.08·4-s + 5-s − 2.25·6-s + 2.92·7-s + 2.44·8-s + 9-s + 2.25·10-s − 5.75·11-s − 3.08·12-s − 1.93·13-s + 6.59·14-s − 15-s − 0.651·16-s − 7.85·17-s + 2.25·18-s + 6.58·19-s + 3.08·20-s − 2.92·21-s − 12.9·22-s − 5.48·23-s − 2.44·24-s + 25-s − 4.36·26-s − 27-s + 9.02·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s − 0.577·3-s + 1.54·4-s + 0.447·5-s − 0.920·6-s + 1.10·7-s + 0.865·8-s + 0.333·9-s + 0.713·10-s − 1.73·11-s − 0.890·12-s − 0.537·13-s + 1.76·14-s − 0.258·15-s − 0.162·16-s − 1.90·17-s + 0.531·18-s + 1.51·19-s + 0.689·20-s − 0.638·21-s − 2.76·22-s − 1.14·23-s − 0.499·24-s + 0.200·25-s − 0.856·26-s − 0.192·27-s + 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 2.25T + 2T^{2} \) |
| 7 | \( 1 - 2.92T + 7T^{2} \) |
| 11 | \( 1 + 5.75T + 11T^{2} \) |
| 13 | \( 1 + 1.93T + 13T^{2} \) |
| 17 | \( 1 + 7.85T + 17T^{2} \) |
| 19 | \( 1 - 6.58T + 19T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 + 5.49T + 31T^{2} \) |
| 37 | \( 1 + 9.97T + 37T^{2} \) |
| 41 | \( 1 + 6.85T + 41T^{2} \) |
| 43 | \( 1 + 2.20T + 43T^{2} \) |
| 47 | \( 1 - 8.90T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 1.87T + 59T^{2} \) |
| 61 | \( 1 - 4.28T + 61T^{2} \) |
| 67 | \( 1 + 5.64T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 - 3.64T + 73T^{2} \) |
| 79 | \( 1 - 3.60T + 79T^{2} \) |
| 83 | \( 1 + 6.09T + 83T^{2} \) |
| 89 | \( 1 - 3.21T + 89T^{2} \) |
| 97 | \( 1 - 18.8T + 97T^{2} \) |
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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
−7.28985070195941285708542721401, −6.99874368041266956313804004051, −5.81507227666377481642978402745, −5.44832283389318147993402112080, −4.92890975501276337375702265426, −4.38676748602014390066563500653, −3.37623751757668170891345538298, −2.33744918568722964496101245047, −1.88651794088964973879873228760, 0,
1.88651794088964973879873228760, 2.33744918568722964496101245047, 3.37623751757668170891345538298, 4.38676748602014390066563500653, 4.92890975501276337375702265426, 5.44832283389318147993402112080, 5.81507227666377481642978402745, 6.99874368041266956313804004051, 7.28985070195941285708542721401
