| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s + 9-s − 3·11-s + 12-s + 13-s − 2·14-s + 16-s − 7·17-s + 18-s − 2·21-s − 3·22-s + 4·23-s + 24-s + 26-s + 27-s − 2·28-s − 29-s + 7·31-s + 32-s − 3·33-s − 7·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s + 0.288·12-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 1.69·17-s + 0.235·18-s − 0.436·21-s − 0.639·22-s + 0.834·23-s + 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.377·28-s − 0.185·29-s + 1.25·31-s + 0.176·32-s − 0.522·33-s − 1.20·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.240207235\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.240207235\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.23931178528942, −13.76750322923106, −13.33022418034979, −12.96798454753562, −12.66594931147037, −11.93172671460768, −11.19502942201528, −11.00262447885586, −10.25412143534457, −9.804049756701376, −9.205220724913643, −8.640699972733049, −8.139780659915608, −7.494620434466705, −6.954723874915668, −6.345190187237545, −6.060470483868051, −5.109446280597079, −4.627852044721486, −4.166905175093165, −3.272923015933079, −2.923121584418626, −2.347866856787822, −1.599805252269958, −0.5037721921758936,
0.5037721921758936, 1.599805252269958, 2.347866856787822, 2.923121584418626, 3.272923015933079, 4.166905175093165, 4.627852044721486, 5.109446280597079, 6.060470483868051, 6.345190187237545, 6.954723874915668, 7.494620434466705, 8.139780659915608, 8.640699972733049, 9.205220724913643, 9.804049756701376, 10.25412143534457, 11.00262447885586, 11.19502942201528, 11.93172671460768, 12.66594931147037, 12.96798454753562, 13.33022418034979, 13.76750322923106, 14.23931178528942
