| L(s) = 1 | − 0.611·2-s − 1.62·4-s − 0.834·5-s − 0.685·7-s + 2.21·8-s + 0.510·10-s + 4.30·11-s − 6.93·13-s + 0.419·14-s + 1.89·16-s − 4.35·17-s + 5.00·19-s + 1.35·20-s − 2.63·22-s + 23-s − 4.30·25-s + 4.24·26-s + 1.11·28-s − 29-s + 9.44·31-s − 5.59·32-s + 2.66·34-s + 0.572·35-s + 7.15·37-s − 3.06·38-s − 1.85·40-s + 3.83·41-s + ⋯ |
| L(s) = 1 | − 0.432·2-s − 0.813·4-s − 0.373·5-s − 0.259·7-s + 0.783·8-s + 0.161·10-s + 1.29·11-s − 1.92·13-s + 0.112·14-s + 0.474·16-s − 1.05·17-s + 1.14·19-s + 0.303·20-s − 0.561·22-s + 0.208·23-s − 0.860·25-s + 0.831·26-s + 0.210·28-s − 0.185·29-s + 1.69·31-s − 0.988·32-s + 0.456·34-s + 0.0967·35-s + 1.17·37-s − 0.496·38-s − 0.292·40-s + 0.599·41-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 | \( 1 + 0.611T + 2T^{2} \) |
| 5 | \( 1 + 0.834T + 5T^{2} \) |
| 7 | \( 1 + 0.685T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 + 6.93T + 13T^{2} \) |
| 17 | \( 1 + 4.35T + 17T^{2} \) |
| 19 | \( 1 - 5.00T + 19T^{2} \) |
| 31 | \( 1 - 9.44T + 31T^{2} \) |
| 37 | \( 1 - 7.15T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 + 7.88T + 43T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 - 5.81T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 14.2T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 - 9.86T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 0.255T + 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 5.84T + 97T^{2} \) |
| 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
−7.917809607652377522195900932631, −7.04190072877200131413349878536, −6.55837278136367246090194361521, −5.43583873380101417782104861196, −4.68182739599019149624575968825, −4.20879325813437091506537288804, −3.29689591158343925653426688929, −2.25998822923146979803712789926, −1.05663978506363017765594773771, 0,
1.05663978506363017765594773771, 2.25998822923146979803712789926, 3.29689591158343925653426688929, 4.20879325813437091506537288804, 4.68182739599019149624575968825, 5.43583873380101417782104861196, 6.55837278136367246090194361521, 7.04190072877200131413349878536, 7.917809607652377522195900932631
