| L(s) = 1 | − 1.96·2-s + 1.84·4-s − 2.84·5-s − 2.21·7-s + 0.305·8-s + 5.57·10-s + 1.29·11-s − 0.673·13-s + 4.34·14-s − 4.28·16-s + 1.19·17-s − 5.61·19-s − 5.24·20-s − 2.54·22-s + 23-s + 3.07·25-s + 1.32·26-s − 4.08·28-s + 29-s − 4.05·31-s + 7.79·32-s − 2.33·34-s + 6.29·35-s + 3.39·37-s + 11.0·38-s − 0.867·40-s + 9.19·41-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.922·4-s − 1.27·5-s − 0.837·7-s + 0.107·8-s + 1.76·10-s + 0.390·11-s − 0.186·13-s + 1.16·14-s − 1.07·16-s + 0.289·17-s − 1.28·19-s − 1.17·20-s − 0.542·22-s + 0.208·23-s + 0.614·25-s + 0.259·26-s − 0.771·28-s + 0.185·29-s − 0.728·31-s + 1.37·32-s − 0.401·34-s + 1.06·35-s + 0.557·37-s + 1.78·38-s − 0.137·40-s + 1.43·41-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 | \( 1 + 1.96T + 2T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 7 | \( 1 + 2.21T + 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 + 0.673T + 13T^{2} \) |
| 17 | \( 1 - 1.19T + 17T^{2} \) |
| 19 | \( 1 + 5.61T + 19T^{2} \) |
| 31 | \( 1 + 4.05T + 31T^{2} \) |
| 37 | \( 1 - 3.39T + 37T^{2} \) |
| 41 | \( 1 - 9.19T + 41T^{2} \) |
| 43 | \( 1 - 9.58T + 43T^{2} \) |
| 47 | \( 1 + 4.53T + 47T^{2} \) |
| 53 | \( 1 - 3.31T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 4.14T + 67T^{2} \) |
| 71 | \( 1 + 4.40T + 71T^{2} \) |
| 73 | \( 1 + 6.12T + 73T^{2} \) |
| 79 | \( 1 - 1.11T + 79T^{2} \) |
| 83 | \( 1 + 2.93T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 0.608T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.77123881221988626054548975267, −7.33389524340374074391613576008, −6.61761669436921476110093940956, −5.88446645814106297942655438109, −4.54932419837864365820321401339, −4.06281076510238537235980520494, −3.13373952182232621054014554023, −2.12172605422210244851273045718, −0.857962473595032970441260868976, 0,
0.857962473595032970441260868976, 2.12172605422210244851273045718, 3.13373952182232621054014554023, 4.06281076510238537235980520494, 4.54932419837864365820321401339, 5.88446645814106297942655438109, 6.61761669436921476110093940956, 7.33389524340374074391613576008, 7.77123881221988626054548975267
