| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 12-s − 2·13-s + 14-s − 15-s + 16-s + 6·17-s + 18-s − 8·19-s − 20-s + 21-s + 24-s + 25-s − 2·26-s + 27-s + 28-s − 6·29-s − 30-s + 4·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s + 0.204·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.182·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.642613228\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.642613228\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.77201435452826, −12.32373440862710, −11.90792025004546, −11.47928118519555, −10.93505761249230, −10.45877286907502, −10.03835433953083, −9.591867937190017, −8.920970817859511, −8.398570704764232, −8.106890551610592, −7.532387395856807, −7.220391507145282, −6.574449010370056, −6.147784231372665, −5.474469836946587, −5.044808933390375, −4.537995436884652, −3.933608179180757, −3.663733860477211, −3.018364057073324, −2.392854886208663, −1.985952722461896, −1.284376806852867, −0.4790899118917553,
0.4790899118917553, 1.284376806852867, 1.985952722461896, 2.392854886208663, 3.018364057073324, 3.663733860477211, 3.933608179180757, 4.537995436884652, 5.044808933390375, 5.474469836946587, 6.147784231372665, 6.574449010370056, 7.220391507145282, 7.532387395856807, 8.106890551610592, 8.398570704764232, 8.920970817859511, 9.591867937190017, 10.03835433953083, 10.45877286907502, 10.93505761249230, 11.47928118519555, 11.90792025004546, 12.32373440862710, 12.77201435452826
