| L(s) = 1 | + 1.03i·5-s + 0.267i·7-s − 3.86·11-s − 2.46·13-s + 6.69i·17-s + 1.73i·19-s − 5.93·23-s + 3.92·25-s + 2.07i·29-s − 0.535i·31-s − 0.277·35-s − 6.46·37-s + 2.07i·41-s − 7.46i·43-s − 9.52·47-s + ⋯ |
| L(s) = 1 | + 0.462i·5-s + 0.101i·7-s − 1.16·11-s − 0.683·13-s + 1.62i·17-s + 0.397i·19-s − 1.23·23-s + 0.785·25-s + 0.384i·29-s − 0.0962i·31-s − 0.0468·35-s − 1.06·37-s + 0.323i·41-s − 1.13i·43-s − 1.38·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.287808 + 0.694831i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.287808 + 0.694831i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.03iT - 5T^{2} \) |
| 7 | \( 1 - 0.267iT - 7T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 + 2.46T + 13T^{2} \) |
| 17 | \( 1 - 6.69iT - 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 - 2.07iT - 29T^{2} \) |
| 31 | \( 1 + 0.535iT - 31T^{2} \) |
| 37 | \( 1 + 6.46T + 37T^{2} \) |
| 41 | \( 1 - 2.07iT - 41T^{2} \) |
| 43 | \( 1 + 7.46iT - 43T^{2} \) |
| 47 | \( 1 + 9.52T + 47T^{2} \) |
| 53 | \( 1 - 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 7.45T + 59T^{2} \) |
| 61 | \( 1 + 9.39T + 61T^{2} \) |
| 67 | \( 1 - 9.73iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 9.92T + 73T^{2} \) |
| 79 | \( 1 - 15.1iT - 79T^{2} \) |
| 83 | \( 1 + 7.72T + 83T^{2} \) |
| 89 | \( 1 - 6.69iT - 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−10.45098744941138433947883460453, −9.884990808803224054028926282396, −8.645497817451599605190804926553, −7.972164347979550472766996351405, −7.11776747924336149030192971092, −6.10516678515247511607458976702, −5.29205515761440934974178865218, −4.13945554139826556990152162582, −3.01257550723240714869896775119, −1.90146857215137225866210874258,
0.33855757848511258435699231339, 2.18956495968578934124179996705, 3.24773642087348178613346681719, 4.78620394190172339872021300690, 5.12263817261008372500312258419, 6.41787837298754772653470932594, 7.42957883265591757776550902338, 8.057644464817822021686736675687, 9.091183279388784477540743677414, 9.826220870506009933117625785723
