| L(s) = 1 | + 5·5-s + 20·7-s + 56·11-s − 86·13-s + 106·17-s + 4·19-s − 136·23-s + 25·25-s + 206·29-s − 152·31-s + 100·35-s + 282·37-s + 246·41-s + 412·43-s − 40·47-s + 57·49-s + 126·53-s + 280·55-s − 56·59-s − 2·61-s − 430·65-s − 388·67-s + 672·71-s + 1.17e3·73-s + 1.12e3·77-s + 408·79-s − 668·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.07·7-s + 1.53·11-s − 1.83·13-s + 1.51·17-s + 0.0482·19-s − 1.23·23-s + 1/5·25-s + 1.31·29-s − 0.880·31-s + 0.482·35-s + 1.25·37-s + 0.937·41-s + 1.46·43-s − 0.124·47-s + 0.166·49-s + 0.326·53-s + 0.686·55-s − 0.123·59-s − 0.00419·61-s − 0.820·65-s − 0.707·67-s + 1.12·71-s + 1.87·73-s + 1.65·77-s + 0.581·79-s − 0.883·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.436999797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.436999797\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 56 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 - 106 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 + 136 T + p^{3} T^{2} \) |
| 29 | \( 1 - 206 T + p^{3} T^{2} \) |
| 31 | \( 1 + 152 T + p^{3} T^{2} \) |
| 37 | \( 1 - 282 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 + 40 T + p^{3} T^{2} \) |
| 53 | \( 1 - 126 T + p^{3} T^{2} \) |
| 59 | \( 1 + 56 T + p^{3} T^{2} \) |
| 61 | \( 1 + 2 T + p^{3} T^{2} \) |
| 67 | \( 1 + 388 T + p^{3} T^{2} \) |
| 71 | \( 1 - 672 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1170 T + p^{3} T^{2} \) |
| 79 | \( 1 - 408 T + p^{3} T^{2} \) |
| 83 | \( 1 + 668 T + p^{3} T^{2} \) |
| 89 | \( 1 + 66 T + p^{3} T^{2} \) |
| 97 | \( 1 + 926 T + p^{3} T^{2} \) |
| 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
−11.08060463809057173683225903507, −9.907583505364535156893006459419, −9.385063965547308619466770471238, −8.095963466080275981182010188311, −7.34476429394049956033475641872, −6.12009820384495302169826102937, −5.07500333046524403748901061041, −4.06076251240465023126578071044, −2.41066860910839893363099644210, −1.14948612848777959996010018860,
1.14948612848777959996010018860, 2.41066860910839893363099644210, 4.06076251240465023126578071044, 5.07500333046524403748901061041, 6.12009820384495302169826102937, 7.34476429394049956033475641872, 8.095963466080275981182010188311, 9.385063965547308619466770471238, 9.907583505364535156893006459419, 11.08060463809057173683225903507
