| L(s) = 1 | + 1.23·3-s − 5-s − 0.381·7-s − 1.47·9-s + 0.618·13-s − 1.23·15-s − 0.763·17-s − 7.09·19-s − 0.472·21-s − 6.85·23-s + 25-s − 5.52·27-s + 3.23·29-s + 2.76·31-s + 0.381·35-s + 5.85·37-s + 0.763·39-s + 4.85·41-s + 4.76·43-s + 1.47·45-s + 4.32·47-s − 6.85·49-s − 0.944·51-s + 12.0·53-s − 8.76·57-s − 4.61·59-s − 8.94·61-s + ⋯ |
| L(s) = 1 | + 0.713·3-s − 0.447·5-s − 0.144·7-s − 0.490·9-s + 0.171·13-s − 0.319·15-s − 0.185·17-s − 1.62·19-s − 0.103·21-s − 1.42·23-s + 0.200·25-s − 1.06·27-s + 0.600·29-s + 0.496·31-s + 0.0645·35-s + 0.962·37-s + 0.122·39-s + 0.758·41-s + 0.726·43-s + 0.219·45-s + 0.631·47-s − 0.979·49-s − 0.132·51-s + 1.66·53-s − 1.16·57-s − 0.601·59-s − 1.14·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.655063559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.655063559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 1.23T + 3T^{2} \) |
| 7 | \( 1 + 0.381T + 7T^{2} \) |
| 13 | \( 1 - 0.618T + 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 + 7.09T + 19T^{2} \) |
| 23 | \( 1 + 6.85T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 - 2.76T + 31T^{2} \) |
| 37 | \( 1 - 5.85T + 37T^{2} \) |
| 41 | \( 1 - 4.85T + 41T^{2} \) |
| 43 | \( 1 - 4.76T + 43T^{2} \) |
| 47 | \( 1 - 4.32T + 47T^{2} \) |
| 53 | \( 1 - 12.0T + 53T^{2} \) |
| 59 | \( 1 + 4.61T + 59T^{2} \) |
| 61 | \( 1 + 8.94T + 61T^{2} \) |
| 67 | \( 1 + 5.23T + 67T^{2} \) |
| 71 | \( 1 + 8.76T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 2.29T + 83T^{2} \) |
| 89 | \( 1 - 17.0T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.894041900003607680192439493765, −7.12037534030346037843643935690, −6.11967973081497228135437846900, −5.96729495272783120044679637259, −4.64610907227142247750983758100, −4.16967067508240579086604271544, −3.41658277652481328852569102095, −2.57241130377826461363688696251, −1.99370046213876412241361982658, −0.56191008818192870719012523499,
0.56191008818192870719012523499, 1.99370046213876412241361982658, 2.57241130377826461363688696251, 3.41658277652481328852569102095, 4.16967067508240579086604271544, 4.64610907227142247750983758100, 5.96729495272783120044679637259, 6.11967973081497228135437846900, 7.12037534030346037843643935690, 7.894041900003607680192439493765
