| L(s) = 1 | + 0.748·2-s + 3-s − 1.43·4-s − 5-s + 0.748·6-s − 7-s − 2.57·8-s + 9-s − 0.748·10-s − 2.94·11-s − 1.43·12-s + 4.64·13-s − 0.748·14-s − 15-s + 0.952·16-s − 5.06·17-s + 0.748·18-s + 5.82·19-s + 1.43·20-s − 21-s − 2.20·22-s − 23-s − 2.57·24-s + 25-s + 3.47·26-s + 27-s + 1.43·28-s + ⋯ |
| L(s) = 1 | + 0.529·2-s + 0.577·3-s − 0.719·4-s − 0.447·5-s + 0.305·6-s − 0.377·7-s − 0.910·8-s + 0.333·9-s − 0.236·10-s − 0.886·11-s − 0.415·12-s + 1.28·13-s − 0.200·14-s − 0.258·15-s + 0.238·16-s − 1.22·17-s + 0.176·18-s + 1.33·19-s + 0.321·20-s − 0.218·21-s − 0.469·22-s − 0.208·23-s − 0.525·24-s + 0.200·25-s + 0.681·26-s + 0.192·27-s + 0.272·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.882367650\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.882367650\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.748T + 2T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 + 5.06T + 17T^{2} \) |
| 19 | \( 1 - 5.82T + 19T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 - 3.55T + 31T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 + 8.28T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 3.00T + 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 - 1.71T + 59T^{2} \) |
| 61 | \( 1 - 8.13T + 61T^{2} \) |
| 67 | \( 1 - 8.33T + 67T^{2} \) |
| 71 | \( 1 - 4.23T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 + 2.47T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 0.696T + 97T^{2} \) |
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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
−8.846651837214603381678600712464, −8.311209707090459449854675012571, −7.55750573146027652864446168921, −6.55237708767310575177857328070, −5.73147896723074094446067238070, −4.85169198945846025464074386379, −4.03799031147446999402801864326, −3.36869340363604525757662313121, −2.50850084898208897096175771783, −0.791412800659401963946366838703,
0.791412800659401963946366838703, 2.50850084898208897096175771783, 3.36869340363604525757662313121, 4.03799031147446999402801864326, 4.85169198945846025464074386379, 5.73147896723074094446067238070, 6.55237708767310575177857328070, 7.55750573146027652864446168921, 8.311209707090459449854675012571, 8.846651837214603381678600712464
