| L(s) = 1 | − 0.498·2-s + 3-s − 1.75·4-s + 0.0200·5-s − 0.498·6-s − 2.67·7-s + 1.87·8-s + 9-s − 0.0100·10-s + 0.400·11-s − 1.75·12-s + 1.83·13-s + 1.33·14-s + 0.0200·15-s + 2.56·16-s − 17-s − 0.498·18-s − 2.84·19-s − 0.0351·20-s − 2.67·21-s − 0.199·22-s + 6.13·23-s + 1.87·24-s − 4.99·25-s − 0.916·26-s + 27-s + 4.67·28-s + ⋯ |
| L(s) = 1 | − 0.352·2-s + 0.577·3-s − 0.875·4-s + 0.00896·5-s − 0.203·6-s − 1.00·7-s + 0.661·8-s + 0.333·9-s − 0.00316·10-s + 0.120·11-s − 0.505·12-s + 0.509·13-s + 0.356·14-s + 0.00517·15-s + 0.642·16-s − 0.242·17-s − 0.117·18-s − 0.652·19-s − 0.00785·20-s − 0.582·21-s − 0.0425·22-s + 1.27·23-s + 0.382·24-s − 0.999·25-s − 0.179·26-s + 0.192·27-s + 0.883·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 + 0.498T + 2T^{2} \) |
| 5 | \( 1 - 0.0200T + 5T^{2} \) |
| 7 | \( 1 + 2.67T + 7T^{2} \) |
| 11 | \( 1 - 0.400T + 11T^{2} \) |
| 13 | \( 1 - 1.83T + 13T^{2} \) |
| 19 | \( 1 + 2.84T + 19T^{2} \) |
| 23 | \( 1 - 6.13T + 23T^{2} \) |
| 29 | \( 1 + 3.39T + 29T^{2} \) |
| 31 | \( 1 + 4.05T + 31T^{2} \) |
| 37 | \( 1 - 6.09T + 37T^{2} \) |
| 41 | \( 1 - 8.26T + 41T^{2} \) |
| 43 | \( 1 + 0.679T + 43T^{2} \) |
| 47 | \( 1 + 5.43T + 47T^{2} \) |
| 53 | \( 1 - 9.14T + 53T^{2} \) |
| 59 | \( 1 + 9.11T + 59T^{2} \) |
| 61 | \( 1 + 5.01T + 61T^{2} \) |
| 67 | \( 1 - 4.00T + 67T^{2} \) |
| 71 | \( 1 - 0.282T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 8.64T + 83T^{2} \) |
| 89 | \( 1 - 4.48T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
| 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
−7.68813579742520110640655136949, −6.88124600421850433571212995677, −6.16888163261313489318828926105, −5.41458579800867545575326971018, −4.47430018383654312410761396680, −3.84527025265927859447146909286, −3.23580117208709954671382887438, −2.23271758414818087893352138894, −1.12592063727972048621488811972, 0,
1.12592063727972048621488811972, 2.23271758414818087893352138894, 3.23580117208709954671382887438, 3.84527025265927859447146909286, 4.47430018383654312410761396680, 5.41458579800867545575326971018, 6.16888163261313489318828926105, 6.88124600421850433571212995677, 7.68813579742520110640655136949
