L(s) = 1 | − 7·5-s + 11·7-s − 3·11-s + 11.3i·13-s + 17·17-s − 19·19-s − 2·23-s + 24·25-s − 39.5i·29-s − 5.65i·31-s − 77·35-s + 39.5i·37-s + 39.5i·41-s − 21·43-s + 5·47-s + ⋯ |
L(s) = 1 | − 1.40·5-s + 1.57·7-s − 0.272·11-s + 0.870i·13-s + 17-s − 19-s − 0.0869·23-s + 0.959·25-s − 1.36i·29-s − 0.182i·31-s − 2.19·35-s + 1.07i·37-s + 0.965i·41-s − 0.488·43-s + 0.106·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.253396881\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.253396881\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 19T \) |
good | 5 | \( 1 + 7T + 25T^{2} \) |
| 7 | \( 1 - 11T + 49T^{2} \) |
| 11 | \( 1 + 3T + 121T^{2} \) |
| 13 | \( 1 - 11.3iT - 169T^{2} \) |
| 17 | \( 1 - 17T + 289T^{2} \) |
| 23 | \( 1 + 2T + 529T^{2} \) |
| 29 | \( 1 + 39.5iT - 841T^{2} \) |
| 31 | \( 1 + 5.65iT - 961T^{2} \) |
| 37 | \( 1 - 39.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 39.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 21T + 1.84e3T^{2} \) |
| 47 | \( 1 - 5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 5.65iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 33.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 23T + 3.72e3T^{2} \) |
| 67 | \( 1 + 39.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 90.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 39T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 118. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 169. iT - 9.40e3T^{2} \) |
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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
−9.579471899502245359051133658681, −8.334390976379347207328400380183, −8.151532739750829528859180034958, −7.43061710651208913548008776354, −6.42119120180256845599308192195, −5.18798226344042811127633184941, −4.42479742106682676182510422101, −3.81297874915072946765816737683, −2.38986715855665952531074239790, −1.14005758116268903527423945457,
0.40844956299243013454826130448, 1.74298061695695176831058000434, 3.16968371537562519942490384499, 4.07577873662142852944064485427, 4.92343680501456391703904393159, 5.64280599633585729859919575834, 7.08593979365430745137786953972, 7.73429533160774630652618919763, 8.240758358445587314196312162055, 8.860014420994767024510751248918