| L(s) = 1 | − 1.16·2-s − 3·3-s − 6.64·4-s + 3.49·6-s − 19.4·7-s + 17.0·8-s + 9·9-s − 11·11-s + 19.9·12-s + 8.26·13-s + 22.6·14-s + 33.3·16-s − 5.66·17-s − 10.4·18-s − 24.4·19-s + 58.4·21-s + 12.7·22-s − 15.3·23-s − 51.1·24-s − 9.61·26-s − 27·27-s + 129.·28-s + 158.·29-s + 302.·31-s − 175.·32-s + 33·33-s + 6.58·34-s + ⋯ |
| L(s) = 1 | − 0.411·2-s − 0.577·3-s − 0.830·4-s + 0.237·6-s − 1.05·7-s + 0.753·8-s + 0.333·9-s − 0.301·11-s + 0.479·12-s + 0.176·13-s + 0.432·14-s + 0.521·16-s − 0.0808·17-s − 0.137·18-s − 0.295·19-s + 0.607·21-s + 0.124·22-s − 0.139·23-s − 0.434·24-s − 0.0725·26-s − 0.192·27-s + 0.873·28-s + 1.01·29-s + 1.75·31-s − 0.967·32-s + 0.174·33-s + 0.0332·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 1.16T + 8T^{2} \) |
| 7 | \( 1 + 19.4T + 343T^{2} \) |
| 13 | \( 1 - 8.26T + 2.19e3T^{2} \) |
| 17 | \( 1 + 5.66T + 4.91e3T^{2} \) |
| 19 | \( 1 + 24.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 15.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 158.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 302.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 266.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 81.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 22.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 15.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 453.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 292.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 255.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 314.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 238.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 744.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 177.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 624.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 342.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.626215678211422750735156769979, −8.608504732146685246665398630774, −7.85528446499196223287206546087, −6.70414963484548336235113733082, −5.98697730590259835567579355211, −4.87919147297819387226592939534, −4.05912712357144854344556389279, −2.80777777735093312923367224897, −1.04015756074746883813924371273, 0,
1.04015756074746883813924371273, 2.80777777735093312923367224897, 4.05912712357144854344556389279, 4.87919147297819387226592939534, 5.98697730590259835567579355211, 6.70414963484548336235113733082, 7.85528446499196223287206546087, 8.608504732146685246665398630774, 9.626215678211422750735156769979
