| L(s) = 1 | − 2.12·2-s + 2.77·3-s + 2.51·4-s − 3.45·5-s − 5.89·6-s + 3.25·7-s − 1.08·8-s + 4.69·9-s + 7.34·10-s − 5.33·11-s + 6.96·12-s + 2.17·13-s − 6.90·14-s − 9.58·15-s − 2.71·16-s + 17-s − 9.96·18-s + 2.09·19-s − 8.67·20-s + 9.02·21-s + 11.3·22-s − 5.48·23-s − 3.00·24-s + 6.94·25-s − 4.60·26-s + 4.69·27-s + 8.16·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 1.60·3-s + 1.25·4-s − 1.54·5-s − 2.40·6-s + 1.22·7-s − 0.383·8-s + 1.56·9-s + 2.32·10-s − 1.60·11-s + 2.01·12-s + 0.601·13-s − 1.84·14-s − 2.47·15-s − 0.679·16-s + 0.242·17-s − 2.34·18-s + 0.480·19-s − 1.94·20-s + 1.96·21-s + 2.41·22-s − 1.14·23-s − 0.613·24-s + 1.38·25-s − 0.904·26-s + 0.904·27-s + 1.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
| good | 2 | \( 1 + 2.12T + 2T^{2} \) |
| 3 | \( 1 - 2.77T + 3T^{2} \) |
| 5 | \( 1 + 3.45T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 11 | \( 1 + 5.33T + 11T^{2} \) |
| 13 | \( 1 - 2.17T + 13T^{2} \) |
| 19 | \( 1 - 2.09T + 19T^{2} \) |
| 23 | \( 1 + 5.48T + 23T^{2} \) |
| 29 | \( 1 + 2.09T + 29T^{2} \) |
| 31 | \( 1 + 0.138T + 31T^{2} \) |
| 37 | \( 1 + 5.49T + 37T^{2} \) |
| 41 | \( 1 + 0.386T + 41T^{2} \) |
| 43 | \( 1 - 4.98T + 43T^{2} \) |
| 47 | \( 1 - 8.26T + 47T^{2} \) |
| 53 | \( 1 + 3.34T + 53T^{2} \) |
| 59 | \( 1 - 0.787T + 59T^{2} \) |
| 61 | \( 1 + 3.36T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 1.66T + 79T^{2} \) |
| 83 | \( 1 + 7.83T + 83T^{2} \) |
| 89 | \( 1 + 2.30T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
| 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
−8.086629182635846967102720059935, −7.51614689603650210532413865074, −7.13979204570204675791927037932, −5.55493853590322009144309027907, −4.52189723577309019592174540193, −3.90067683209514260062093052452, −2.97893197082237061979166734019, −2.18958735988400021701792600048, −1.30199688695225429466737454016, 0,
1.30199688695225429466737454016, 2.18958735988400021701792600048, 2.97893197082237061979166734019, 3.90067683209514260062093052452, 4.52189723577309019592174540193, 5.55493853590322009144309027907, 7.13979204570204675791927037932, 7.51614689603650210532413865074, 8.086629182635846967102720059935
