| 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
| \(L(1)\) |
\(\approx\) |
\(1.480835460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.480835460\) |
| \(L(\frac{3}{2})\) |
|
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
−9.733548647403556288003391945072, −8.706104461463539807303592850178, −7.74134752980582245354121903439, −7.00177847821925514379884772079, −6.20360738934172181959288944086, −5.76173204259112309199558448335, −4.22291253318670711835678525572, −3.29394274068669346173671914314, −2.84995445571011989286205007998, −1.02421974642560016056953714681,
0.72632800510195580356645562087, 2.09033972609824944502991616649, 3.35718497285526214946854238863, 4.30246165220481785405691954981, 5.02727162060997566780395037111, 6.08178654570327375627459569565, 7.08152485267834674868686822925, 7.34071734221558925607189572278, 8.759268014159585718278129866737, 9.284753817202094820022056387526
