| L(s) = 1 | + 2-s + 4-s − 2.41·5-s − 3·7-s + 8-s − 2.41·10-s + 2.82·11-s + 1.17·13-s − 3·14-s + 16-s + 0.828·17-s + 7.65·19-s − 2.41·20-s + 2.82·22-s − 8.82·23-s + 0.828·25-s + 1.17·26-s − 3·28-s − 6.82·29-s − 2.17·31-s + 32-s + 0.828·34-s + 7.24·35-s + 1.24·37-s + 7.65·38-s − 2.41·40-s − 11.6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.07·5-s − 1.13·7-s + 0.353·8-s − 0.763·10-s + 0.852·11-s + 0.324·13-s − 0.801·14-s + 0.250·16-s + 0.200·17-s + 1.75·19-s − 0.539·20-s + 0.603·22-s − 1.84·23-s + 0.165·25-s + 0.229·26-s − 0.566·28-s − 1.26·29-s − 0.390·31-s + 0.176·32-s + 0.142·34-s + 1.22·35-s + 0.204·37-s + 1.24·38-s − 0.381·40-s − 1.82·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2.41T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 - 0.828T + 17T^{2} \) |
| 19 | \( 1 - 7.65T + 19T^{2} \) |
| 23 | \( 1 + 8.82T + 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 - 1.24T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 5.48T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 3.58T + 59T^{2} \) |
| 61 | \( 1 + 8.48T + 61T^{2} \) |
| 67 | \( 1 + 2.41T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 - 4.89T + 83T^{2} \) |
| 89 | \( 1 - 2.65T + 89T^{2} \) |
| 97 | \( 1 + 1.82T + 97T^{2} \) |
| 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
−8.131496850631192603496395117050, −7.50326883287050542137046189949, −6.81128544528354893862414194158, −6.03416497086788620489504511302, −5.33254518163463249478239239730, −4.08385515220788344171074271657, −3.69564130157039186474002141146, −3.02770930942286539656898943206, −1.56493651876189671881872294723, 0,
1.56493651876189671881872294723, 3.02770930942286539656898943206, 3.69564130157039186474002141146, 4.08385515220788344171074271657, 5.33254518163463249478239239730, 6.03416497086788620489504511302, 6.81128544528354893862414194158, 7.50326883287050542137046189949, 8.131496850631192603496395117050
