L(s) = 1 | + i·2-s − 4-s + 1.73·5-s + (2 − 1.73i)7-s − i·8-s + 1.73i·10-s + 3i·11-s − 3.46i·13-s + (1.73 + 2i)14-s + 16-s + 6.92·17-s − 1.73i·19-s − 1.73·20-s − 3·22-s + 3i·23-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.774·5-s + (0.755 − 0.654i)7-s − 0.353i·8-s + 0.547i·10-s + 0.904i·11-s − 0.960i·13-s + (0.462 + 0.534i)14-s + 0.250·16-s + 1.68·17-s − 0.397i·19-s − 0.387·20-s − 0.639·22-s + 0.625i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48551 + 0.553835i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48551 + 0.553835i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 3.46T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 + 3iT - 71T^{2} \) |
| 73 | \( 1 - 3.46iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−11.43916314926851110657862689488, −10.11871327298893430421623478917, −9.896235992477110920352582699871, −8.480156461643615321021244773683, −7.65552021206789772386952219897, −6.84991170379380529438962191151, −5.52125640432607666596241530268, −4.93411973703737854504041244834, −3.41879298526214755288133422425, −1.48579618099667282894445045004,
1.51008882502341006757253193254, 2.71904355305484860998769327284, 4.15556038138719123885405847202, 5.45834006401551436258061214975, 6.12703243987095352421551891844, 7.81229356765347982834276473699, 8.624994435318074747412456594025, 9.604336155420799240544733645748, 10.27185399734461277503975208608, 11.52773240506220881516800536629