| L(s) = 1 | + 2i·5-s + 4.24·7-s + 3·9-s + 4.24i·11-s − i·13-s + 4·17-s − 4.24i·19-s + 25-s − 2i·29-s − 4.24·31-s + 8.48i·35-s + 6i·37-s − 2·41-s − 8.48i·43-s + 6i·45-s + ⋯ |
| L(s) = 1 | + 0.894i·5-s + 1.60·7-s + 9-s + 1.27i·11-s − 0.277i·13-s + 0.970·17-s − 0.973i·19-s + 0.200·25-s − 0.371i·29-s − 0.762·31-s + 1.43i·35-s + 0.986i·37-s − 0.312·41-s − 1.29i·43-s + 0.894i·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.329210415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.329210415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 + 4.24iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8.48iT - 43T^{2} \) |
| 47 | \( 1 + 12.7T + 47T^{2} \) |
| 53 | \( 1 - 8iT - 53T^{2} \) |
| 59 | \( 1 + 4.24iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 4.24iT - 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8.48T + 79T^{2} \) |
| 83 | \( 1 + 12.7iT - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−9.644592681313440543883955534471, −8.572372092713778530834141361318, −7.57447996853827958546500101358, −7.33558403085875822351346917832, −6.42055689499353052851356877084, −5.05642525178979433601922799255, −4.69337878070209477303930715328, −3.54259911015932402957629655639, −2.27726227573889686649907543777, −1.40429199639798297162781704918,
1.09800191967246781836076274332, 1.74611724752208114260128855334, 3.44043280158304858377402309423, 4.36816585228011739741795730131, 5.13901722377133402532740563086, 5.76823907808462301793442007759, 6.99169062147675151487866861707, 8.036935800287916940628743802293, 8.227372549381921243618457196952, 9.166735586869037531998075476945
