| L(s) = 1 | + 2·2-s + 2·4-s − 4·7-s − 3·9-s − 11-s − 2·13-s − 8·14-s − 4·16-s − 2·17-s − 6·18-s − 2·22-s + 6·23-s − 4·26-s − 8·28-s + 9·29-s − 7·31-s − 8·32-s − 4·34-s − 6·36-s + 2·37-s + 2·41-s + 2·43-s − 2·44-s + 12·46-s + 6·47-s + 9·49-s − 4·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.51·7-s − 9-s − 0.301·11-s − 0.554·13-s − 2.13·14-s − 16-s − 0.485·17-s − 1.41·18-s − 0.426·22-s + 1.25·23-s − 0.784·26-s − 1.51·28-s + 1.67·29-s − 1.25·31-s − 1.41·32-s − 0.685·34-s − 36-s + 0.328·37-s + 0.312·41-s + 0.304·43-s − 0.301·44-s + 1.76·46-s + 0.875·47-s + 9/7·49-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.105016517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.105016517\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.34444403405367046839408768692, −6.82504700909223389822141819119, −6.17427872773578038096006877803, −5.64725602868882740424934611950, −4.98882044221344099042286742266, −4.24268102781465038145714028719, −3.39219390130035433443558897600, −2.87385836530491516372746645265, −2.37402654009108351731947703058, −0.52108213209354670262371061679,
0.52108213209354670262371061679, 2.37402654009108351731947703058, 2.87385836530491516372746645265, 3.39219390130035433443558897600, 4.24268102781465038145714028719, 4.98882044221344099042286742266, 5.64725602868882740424934611950, 6.17427872773578038096006877803, 6.82504700909223389822141819119, 7.34444403405367046839408768692
