| L(s) = 1 | − 1.16·2-s − 0.646·4-s + 5-s + 2.23·7-s + 3.07·8-s − 1.16·10-s − 6.24·11-s − 4.63·13-s − 2.59·14-s − 2.28·16-s + 2.75·17-s + 3.29·19-s − 0.646·20-s + 7.26·22-s + 6.86·23-s + 25-s + 5.39·26-s − 1.44·28-s − 1.22·29-s − 5.69·31-s − 3.49·32-s − 3.20·34-s + 2.23·35-s + 6.87·37-s − 3.83·38-s + 3.07·40-s − 0.954·41-s + ⋯ |
| L(s) = 1 | − 0.822·2-s − 0.323·4-s + 0.447·5-s + 0.843·7-s + 1.08·8-s − 0.367·10-s − 1.88·11-s − 1.28·13-s − 0.693·14-s − 0.572·16-s + 0.668·17-s + 0.756·19-s − 0.144·20-s + 1.54·22-s + 1.43·23-s + 0.200·25-s + 1.05·26-s − 0.272·28-s − 0.227·29-s − 1.02·31-s − 0.617·32-s − 0.550·34-s + 0.377·35-s + 1.13·37-s − 0.622·38-s + 0.486·40-s − 0.149·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| good | 2 | \( 1 + 1.16T + 2T^{2} \) |
| 7 | \( 1 - 2.23T + 7T^{2} \) |
| 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 - 2.75T + 17T^{2} \) |
| 19 | \( 1 - 3.29T + 19T^{2} \) |
| 23 | \( 1 - 6.86T + 23T^{2} \) |
| 29 | \( 1 + 1.22T + 29T^{2} \) |
| 31 | \( 1 + 5.69T + 31T^{2} \) |
| 37 | \( 1 - 6.87T + 37T^{2} \) |
| 41 | \( 1 + 0.954T + 41T^{2} \) |
| 43 | \( 1 + 12.5T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 7.06T + 59T^{2} \) |
| 61 | \( 1 + 2.99T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 9.04T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 0.814T + 79T^{2} \) |
| 83 | \( 1 + 2.94T + 83T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
| 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
−7.922138204792025372792991674046, −7.67974358609335487970465094293, −7.01550092155620524129510854830, −5.50974422415752322051854377295, −5.14933737380280410649319451226, −4.60610904335991343843242830432, −3.17007105188439227936852841796, −2.29828788684887928615226345584, −1.26841680487887209092901813962, 0,
1.26841680487887209092901813962, 2.29828788684887928615226345584, 3.17007105188439227936852841796, 4.60610904335991343843242830432, 5.14933737380280410649319451226, 5.50974422415752322051854377295, 7.01550092155620524129510854830, 7.67974358609335487970465094293, 7.922138204792025372792991674046
