| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 2·7-s + 8-s + 9-s − 2·11-s + 12-s + 2·14-s + 16-s + 2·17-s + 18-s + 19-s + 2·21-s − 2·22-s + 8·23-s + 24-s + 27-s + 2·28-s + 32-s − 2·33-s + 2·34-s + 36-s − 4·37-s + 38-s − 8·41-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 − 0.603·11-s + 0.288·12-s + 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.229·19-s + 0.436·21-s − 0.426·22-s + 1.66·23-s + 0.204·24-s + 0.192·27-s + 0.377·28-s + 0.176·32-s − 0.348·33-s + 0.342·34-s + 1/6·36-s − 0.657·37-s + 0.162·38-s − 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.074442845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.074442845\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 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
−8.702486642929141982486881259242, −7.907856463659196044490538529050, −7.33642093609585474040910511555, −6.55595108385726602829957036220, −5.37031935185807141853486458305, −5.01501769681507263453902359061, −3.99706983643862778383960384165, −3.13509852697518272934346209828, −2.31850648944287911229531871988, −1.19691856529568672013010239136,
1.19691856529568672013010239136, 2.31850648944287911229531871988, 3.13509852697518272934346209828, 3.99706983643862778383960384165, 5.01501769681507263453902359061, 5.37031935185807141853486458305, 6.55595108385726602829957036220, 7.33642093609585474040910511555, 7.907856463659196044490538529050, 8.702486642929141982486881259242
