| L(s) = 1 | + 3·3-s − 5-s − 3·7-s + 6·9-s + 5·11-s + 2·13-s − 3·15-s + 4·17-s − 4·19-s − 9·21-s + 6·23-s + 25-s + 9·27-s + 6·29-s − 4·31-s + 15·33-s + 3·35-s − 37-s + 6·39-s − 9·41-s + 10·43-s − 6·45-s − 11·47-s + 2·49-s + 12·51-s − 11·53-s − 5·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s − 1.13·7-s + 2·9-s + 1.50·11-s + 0.554·13-s − 0.774·15-s + 0.970·17-s − 0.917·19-s − 1.96·21-s + 1.25·23-s + 1/5·25-s + 1.73·27-s + 1.11·29-s − 0.718·31-s + 2.61·33-s + 0.507·35-s − 0.164·37-s + 0.960·39-s − 1.40·41-s + 1.52·43-s − 0.894·45-s − 1.60·47-s + 2/7·49-s + 1.68·51-s − 1.51·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.534996595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.534996595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| 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
−10.03174952203399649378141239133, −9.284993923943248760027735739396, −8.798405749374759083599131072327, −7.965299355909049566818828913125, −6.96228910991813633687344598913, −6.31311251034846705906848780783, −4.48878031439686602633803909869, −3.49416389960600763793673657644, −3.08976764586911151057853710056, −1.47616131815811688268284937786,
1.47616131815811688268284937786, 3.08976764586911151057853710056, 3.49416389960600763793673657644, 4.48878031439686602633803909869, 6.31311251034846705906848780783, 6.96228910991813633687344598913, 7.965299355909049566818828913125, 8.798405749374759083599131072327, 9.284993923943248760027735739396, 10.03174952203399649378141239133
