| L(s) = 1 | − 2-s − 1.23·3-s + 4-s + 1.23·6-s − 7-s − 8-s − 1.47·9-s + 11-s − 1.23·12-s + 3.23·13-s + 14-s + 16-s − 2.47·17-s + 1.47·18-s − 7.23·19-s + 1.23·21-s − 22-s − 4·23-s + 1.23·24-s − 3.23·26-s + 5.52·27-s − 28-s + 4.47·29-s + 2·31-s − 32-s − 1.23·33-s + 2.47·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.713·3-s + 0.5·4-s + 0.504·6-s − 0.377·7-s − 0.353·8-s − 0.490·9-s + 0.301·11-s − 0.356·12-s + 0.897·13-s + 0.267·14-s + 0.250·16-s − 0.599·17-s + 0.346·18-s − 1.66·19-s + 0.269·21-s − 0.213·22-s − 0.834·23-s + 0.252·24-s − 0.634·26-s + 1.06·27-s − 0.188·28-s + 0.830·29-s + 0.359·31-s − 0.176·32-s − 0.215·33-s + 0.423·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6161987984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6161987984\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.23T + 3T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 2.76T + 59T^{2} \) |
| 61 | \( 1 + 0.763T + 61T^{2} \) |
| 67 | \( 1 + 11.4T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.684648638717238585121319377801, −7.896314873822675993737841246852, −6.83134186277352371810084411107, −6.26833104785699265051346445140, −5.89549374886094321173671841092, −4.72855063337205151094649498891, −3.87976465877525901154334684993, −2.81501425497379130970789188319, −1.80418779913884958590731653189, −0.51124012064741364046290632354,
0.51124012064741364046290632354, 1.80418779913884958590731653189, 2.81501425497379130970789188319, 3.87976465877525901154334684993, 4.72855063337205151094649498891, 5.89549374886094321173671841092, 6.26833104785699265051346445140, 6.83134186277352371810084411107, 7.896314873822675993737841246852, 8.684648638717238585121319377801
