| L(s) = 1 | + 5·2-s + 17·4-s + 5·5-s + 8·7-s + 45·8-s + 25·10-s + 56·11-s − 13·13-s + 40·14-s + 89·16-s − 58·17-s + 24·19-s + 85·20-s + 280·22-s − 36·23-s + 25·25-s − 65·26-s + 136·28-s + 242·29-s − 64·31-s + 85·32-s − 290·34-s + 40·35-s − 254·37-s + 120·38-s + 225·40-s + 414·41-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 17/8·4-s + 0.447·5-s + 0.431·7-s + 1.98·8-s + 0.790·10-s + 1.53·11-s − 0.277·13-s + 0.763·14-s + 1.39·16-s − 0.827·17-s + 0.289·19-s + 0.950·20-s + 2.71·22-s − 0.326·23-s + 1/5·25-s − 0.490·26-s + 0.917·28-s + 1.54·29-s − 0.370·31-s + 0.469·32-s − 1.46·34-s + 0.193·35-s − 1.12·37-s + 0.512·38-s + 0.889·40-s + 1.57·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.119945977\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.119945977\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 13 | \( 1 + p T \) |
| good | 2 | \( 1 - 5 T + p^{3} T^{2} \) |
| 7 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 56 T + p^{3} T^{2} \) |
| 17 | \( 1 + 58 T + p^{3} T^{2} \) |
| 19 | \( 1 - 24 T + p^{3} T^{2} \) |
| 23 | \( 1 + 36 T + p^{3} T^{2} \) |
| 29 | \( 1 - 242 T + p^{3} T^{2} \) |
| 31 | \( 1 + 64 T + p^{3} T^{2} \) |
| 37 | \( 1 + 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 414 T + p^{3} T^{2} \) |
| 43 | \( 1 + 164 T + p^{3} T^{2} \) |
| 47 | \( 1 - 40 T + p^{3} T^{2} \) |
| 53 | \( 1 + 82 T + p^{3} T^{2} \) |
| 59 | \( 1 - 744 T + p^{3} T^{2} \) |
| 61 | \( 1 - 494 T + p^{3} T^{2} \) |
| 67 | \( 1 + 508 T + p^{3} T^{2} \) |
| 71 | \( 1 + 384 T + p^{3} T^{2} \) |
| 73 | \( 1 - 462 T + p^{3} T^{2} \) |
| 79 | \( 1 + 816 T + p^{3} T^{2} \) |
| 83 | \( 1 - 92 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1210 T + p^{3} T^{2} \) |
| 97 | \( 1 + 530 T + p^{3} T^{2} \) |
| 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
−10.66314462299208068721780379594, −9.524043904720436877241204996493, −8.470572355941060804393688157974, −7.04152889624249031380699304090, −6.49595060497168841007882116210, −5.55089899644257877216491594769, −4.60289478527050021799776391921, −3.85670319069666355110450058196, −2.62748866352326127846126721422, −1.49888841852516352614339280412,
1.49888841852516352614339280412, 2.62748866352326127846126721422, 3.85670319069666355110450058196, 4.60289478527050021799776391921, 5.55089899644257877216491594769, 6.49595060497168841007882116210, 7.04152889624249031380699304090, 8.470572355941060804393688157974, 9.524043904720436877241204996493, 10.66314462299208068721780379594
