| L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s + 3·11-s + 2·14-s + 16-s − 6·17-s + 2·19-s + 20-s + 3·22-s − 3·23-s + 25-s + 2·28-s − 3·29-s + 5·31-s + 32-s − 6·34-s + 2·35-s − 7·37-s + 2·38-s + 40-s − 6·41-s − 43-s + 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s + 0.223·20-s + 0.639·22-s − 0.625·23-s + 1/5·25-s + 0.377·28-s − 0.557·29-s + 0.898·31-s + 0.176·32-s − 1.02·34-s + 0.338·35-s − 1.15·37-s + 0.324·38-s + 0.158·40-s − 0.937·41-s − 0.152·43-s + 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.595593322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.595593322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.76810156596589, −15.51899005630622, −14.76748399006980, −14.36068314896597, −13.73031899032137, −13.48544331951841, −12.73547612614990, −12.03172789868809, −11.63132527937674, −11.11860789248583, −10.48008326735865, −9.834552121554374, −9.143430654085216, −8.545757215917024, −7.961625475512797, −7.070145148706103, −6.610826602811269, −6.067540231970081, −5.135356172203483, −4.856858168713826, −3.940313549436735, −3.477084736550262, −2.223692381534583, −1.954524536629318, −0.8432017549773610,
0.8432017549773610, 1.954524536629318, 2.223692381534583, 3.477084736550262, 3.940313549436735, 4.856858168713826, 5.135356172203483, 6.067540231970081, 6.610826602811269, 7.070145148706103, 7.961625475512797, 8.545757215917024, 9.143430654085216, 9.834552121554374, 10.48008326735865, 11.11860789248583, 11.63132527937674, 12.03172789868809, 12.73547612614990, 13.48544331951841, 13.73031899032137, 14.36068314896597, 14.76748399006980, 15.51899005630622, 15.76810156596589
