| L(s) = 1 | − 1.75·2-s + 1.08·4-s + 2.67·5-s + 0.0258·7-s + 1.60·8-s − 4.70·10-s − 1.36·11-s − 2.10·13-s − 0.0453·14-s − 4.99·16-s + 2.63·17-s − 5.19·19-s + 2.91·20-s + 2.39·22-s + 23-s + 2.16·25-s + 3.70·26-s + 0.0280·28-s − 29-s + 6.67·31-s + 5.57·32-s − 4.63·34-s + 0.0690·35-s − 9.24·37-s + 9.13·38-s + 4.28·40-s + 9.58·41-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.544·4-s + 1.19·5-s + 0.00975·7-s + 0.566·8-s − 1.48·10-s − 0.410·11-s − 0.583·13-s − 0.0121·14-s − 1.24·16-s + 0.639·17-s − 1.19·19-s + 0.651·20-s + 0.510·22-s + 0.208·23-s + 0.432·25-s + 0.725·26-s + 0.00531·28-s − 0.185·29-s + 1.19·31-s + 0.984·32-s − 0.794·34-s + 0.0116·35-s − 1.51·37-s + 1.48·38-s + 0.677·40-s + 1.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.75T + 2T^{2} \) |
| 5 | \( 1 - 2.67T + 5T^{2} \) |
| 7 | \( 1 - 0.0258T + 7T^{2} \) |
| 11 | \( 1 + 1.36T + 11T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 + 5.19T + 19T^{2} \) |
| 31 | \( 1 - 6.67T + 31T^{2} \) |
| 37 | \( 1 + 9.24T + 37T^{2} \) |
| 41 | \( 1 - 9.58T + 41T^{2} \) |
| 43 | \( 1 + 5.52T + 43T^{2} \) |
| 47 | \( 1 - 0.699T + 47T^{2} \) |
| 53 | \( 1 - 0.391T + 53T^{2} \) |
| 59 | \( 1 + 1.26T + 59T^{2} \) |
| 61 | \( 1 - 4.70T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 3.62T + 71T^{2} \) |
| 73 | \( 1 - 1.31T + 73T^{2} \) |
| 79 | \( 1 + 9.67T + 79T^{2} \) |
| 83 | \( 1 + 8.72T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 3.61T + 97T^{2} \) |
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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
−8.011089763832443678610337152020, −7.07746380179747732891927357082, −6.53891129427622837694116374073, −5.63873071432079865698255050825, −4.98809554535718347348600430072, −4.11794426695114849178073181413, −2.79510112681685228576256769038, −2.07354736183039645922763263742, −1.27019285643809201845411216852, 0,
1.27019285643809201845411216852, 2.07354736183039645922763263742, 2.79510112681685228576256769038, 4.11794426695114849178073181413, 4.98809554535718347348600430072, 5.63873071432079865698255050825, 6.53891129427622837694116374073, 7.07746380179747732891927357082, 8.011089763832443678610337152020
