| L(s) = 1 | − 2·2-s − 2·3-s + 4·4-s + 4·6-s − 8·8-s − 23·9-s + 48·11-s − 8·12-s + 56·13-s + 16·16-s − 114·17-s + 46·18-s − 2·19-s − 96·22-s + 120·23-s + 16·24-s − 112·26-s + 100·27-s − 54·29-s − 236·31-s − 32·32-s − 96·33-s + 228·34-s − 92·36-s − 146·37-s + 4·38-s − 112·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.384·3-s + 1/2·4-s + 0.272·6-s − 0.353·8-s − 0.851·9-s + 1.31·11-s − 0.192·12-s + 1.19·13-s + 1/4·16-s − 1.62·17-s + 0.602·18-s − 0.0241·19-s − 0.930·22-s + 1.08·23-s + 0.136·24-s − 0.844·26-s + 0.712·27-s − 0.345·29-s − 1.36·31-s − 0.176·32-s − 0.506·33-s + 1.15·34-s − 0.425·36-s − 0.648·37-s + 0.0170·38-s − 0.459·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 48 T + p^{3} T^{2} \) |
| 13 | \( 1 - 56 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 2 T + p^{3} T^{2} \) |
| 23 | \( 1 - 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 236 T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 376 T + p^{3} T^{2} \) |
| 47 | \( 1 + 12 T + p^{3} T^{2} \) |
| 53 | \( 1 + 174 T + p^{3} T^{2} \) |
| 59 | \( 1 + 138 T + p^{3} T^{2} \) |
| 61 | \( 1 + 380 T + p^{3} T^{2} \) |
| 67 | \( 1 - 484 T + p^{3} T^{2} \) |
| 71 | \( 1 - 576 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1150 T + p^{3} T^{2} \) |
| 79 | \( 1 - 776 T + p^{3} T^{2} \) |
| 83 | \( 1 - 378 T + p^{3} T^{2} \) |
| 89 | \( 1 - 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1330 T + p^{3} T^{2} \) |
| 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
−8.539341592551001387557533948219, −7.39973031258187083286753697521, −6.58202788721291089746538972664, −6.16461554263691471294167052764, −5.20568673328082435683710049104, −4.08385238463036686617696978946, −3.24687628678457876306757118077, −2.04681575153261063494012402412, −1.08141138793158350672486529299, 0,
1.08141138793158350672486529299, 2.04681575153261063494012402412, 3.24687628678457876306757118077, 4.08385238463036686617696978946, 5.20568673328082435683710049104, 6.16461554263691471294167052764, 6.58202788721291089746538972664, 7.39973031258187083286753697521, 8.539341592551001387557533948219
