| L(s) = 1 | + 5-s − 3.56·7-s + 11-s − 3.12·13-s − 5.56·17-s − 2.43·19-s + 7.12·23-s + 25-s + 0.438·29-s − 8.68·31-s − 3.56·35-s + 9.80·37-s + 10·41-s − 5.12·43-s − 7.12·47-s + 5.68·49-s + 4.43·53-s + 55-s − 13.3·59-s − 3.56·61-s − 3.12·65-s − 2.43·71-s + 4.87·73-s − 3.56·77-s − 0.876·79-s + 10·83-s − 5.56·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.34·7-s + 0.301·11-s − 0.866·13-s − 1.34·17-s − 0.559·19-s + 1.48·23-s + 0.200·25-s + 0.0814·29-s − 1.55·31-s − 0.602·35-s + 1.61·37-s + 1.56·41-s − 0.781·43-s − 1.03·47-s + 0.812·49-s + 0.609·53-s + 0.134·55-s − 1.74·59-s − 0.456·61-s − 0.387·65-s − 0.289·71-s + 0.570·73-s − 0.405·77-s − 0.0986·79-s + 1.09·83-s − 0.603·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.258219975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.258219975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + 3.56T + 7T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 + 5.56T + 17T^{2} \) |
| 19 | \( 1 + 2.43T + 19T^{2} \) |
| 23 | \( 1 - 7.12T + 23T^{2} \) |
| 29 | \( 1 - 0.438T + 29T^{2} \) |
| 31 | \( 1 + 8.68T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 + 7.12T + 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 2.43T + 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 + 0.876T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
| 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
−7.65708116697806485349987213814, −7.06337239136741762416601659536, −6.41115178649637387882503982691, −5.99941936414157809903869356021, −4.96735312792841245365505116814, −4.37005447899621997875291863236, −3.38573593399040074281873417345, −2.70246170226029660898107193846, −1.91954721977071906986176674558, −0.53045920787695417442887974006,
0.53045920787695417442887974006, 1.91954721977071906986176674558, 2.70246170226029660898107193846, 3.38573593399040074281873417345, 4.37005447899621997875291863236, 4.96735312792841245365505116814, 5.99941936414157809903869356021, 6.41115178649637387882503982691, 7.06337239136741762416601659536, 7.65708116697806485349987213814
