| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 7-s + 8-s + 9-s + 10-s + 6·11-s + 12-s + 13-s + 14-s + 15-s + 16-s − 17-s + 18-s + 8·19-s + 20-s + 21-s + 6·22-s − 6·23-s + 24-s + 25-s + 26-s + 27-s + 28-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 + 1.80·11-s + 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.83·19-s + 0.223·20-s + 0.218·21-s + 1.27·22-s − 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.494895470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.494895470\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.31818739191803, −14.22327614509370, −13.56184371391259, −13.47429513180473, −12.44750660096618, −12.05539695882001, −11.62037273052798, −11.27196542654731, −10.28366579946496, −9.949561419134629, −9.395396265060346, −8.856576542960562, −8.222847679143717, −7.741640443195931, −6.982793164175322, −6.518125781738914, −6.037010752812928, −5.397990052760057, −4.540472831266390, −4.288368868906024, −3.467341927829151, −3.015583367615290, −2.207068121071194, −1.427279243353953, −1.011408642176661,
1.011408642176661, 1.427279243353953, 2.207068121071194, 3.015583367615290, 3.467341927829151, 4.288368868906024, 4.540472831266390, 5.397990052760057, 6.037010752812928, 6.518125781738914, 6.982793164175322, 7.741640443195931, 8.222847679143717, 8.856576542960562, 9.395396265060346, 9.949561419134629, 10.28366579946496, 11.27196542654731, 11.62037273052798, 12.05539695882001, 12.44750660096618, 13.47429513180473, 13.56184371391259, 14.22327614509370, 14.31818739191803
