| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s + 3·8-s + 9-s − 10-s + 4·11-s − 12-s + 2·13-s + 15-s − 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 4·22-s + 3·24-s + 25-s − 2·26-s + 27-s + 2·29-s − 30-s − 5·32-s + 4·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.883·32-s + 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 41 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−15.68164394642440, −14.89848863551871, −14.61734301190701, −13.88969842821989, −13.61464233004950, −13.12379677875317, −12.41166118757646, −11.93336562795837, −10.97713790470647, −10.69010247357684, −10.02449228050021, −9.407760481585933, −9.071523639302205, −8.545105119038792, −8.161497195449445, −7.377595665832176, −6.615087864771428, −6.356592160396822, −5.310900564854340, −4.702272880546634, −3.960654254335747, −3.581212074483096, −2.500626019742822, −1.686254358439200, −1.186747828672650, 0,
1.186747828672650, 1.686254358439200, 2.500626019742822, 3.581212074483096, 3.960654254335747, 4.702272880546634, 5.310900564854340, 6.356592160396822, 6.615087864771428, 7.377595665832176, 8.161497195449445, 8.545105119038792, 9.071523639302205, 9.407760481585933, 10.02449228050021, 10.69010247357684, 10.97713790470647, 11.93336562795837, 12.41166118757646, 13.12379677875317, 13.61464233004950, 13.88969842821989, 14.61734301190701, 14.89848863551871, 15.68164394642440
