| L(s) = 1 | + 4.38i·5-s + i·7-s + 2.31·11-s − 4.73·13-s + 5.27i·17-s + 5i·19-s + 0.517·23-s − 14.1·25-s − 4.24i·29-s + 3.53i·31-s − 4.38·35-s − 4.46·37-s − 6.45i·41-s + 1.26i·43-s − 8.10·47-s + ⋯ |
| L(s) = 1 | + 1.95i·5-s + 0.377i·7-s + 0.696·11-s − 1.31·13-s + 1.28i·17-s + 1.14i·19-s + 0.107·23-s − 2.83·25-s − 0.787i·29-s + 0.635i·31-s − 0.740·35-s − 0.733·37-s − 1.00i·41-s + 0.193i·43-s − 1.18·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8548949013\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8548949013\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 4.38iT - 5T^{2} \) |
| 11 | \( 1 - 2.31T + 11T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 17 | \( 1 - 5.27iT - 17T^{2} \) |
| 19 | \( 1 - 5iT - 19T^{2} \) |
| 23 | \( 1 - 0.517T + 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 3.53iT - 31T^{2} \) |
| 37 | \( 1 + 4.46T + 37T^{2} \) |
| 41 | \( 1 + 6.45iT - 41T^{2} \) |
| 43 | \( 1 - 1.26iT - 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 + 5.93iT - 53T^{2} \) |
| 59 | \( 1 + 6.31T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 9.66iT - 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 17.6T + 83T^{2} \) |
| 89 | \( 1 - 0.517iT - 89T^{2} \) |
| 97 | \( 1 - 0.928T + 97T^{2} \) |
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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
−8.301707632003993182309709566239, −7.73084395134079075591686323295, −7.01634478329074750255753367130, −6.42503676233018746864611147775, −5.94695473769334744502876088714, −4.97795678840451598657634167899, −3.77639046749902164265214279759, −3.44749787156701084663076288907, −2.38337746568990344065629523210, −1.82855553907130628820850649175,
0.22948232768961043615105816017, 1.03518939179099225735620068217, 2.02797123380426936886867699404, 3.13973349923110398369655981234, 4.25596404548046373838049239114, 4.87121533097530527517568815549, 5.09211252208485966549059008505, 6.14570334585667639954395787424, 7.13670424764229698127630586413, 7.56247757769646554023489124840
