| L(s) = 1 | + (−3 − 4.24i)3-s − 5.65i·5-s + 16.9i·7-s + (−8.99 + 25.4i)9-s − 30·11-s − 72·13-s + (−24 + 16.9i)15-s + 50.9i·17-s − 25.4i·19-s + (71.9 − 50.9i)21-s + 144·23-s + 93·25-s + (134. − 38.1i)27-s + 5.65i·29-s − 220. i·31-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.816i)3-s − 0.505i·5-s + 0.916i·7-s + (−0.333 + 0.942i)9-s − 0.822·11-s − 1.53·13-s + (−0.413 + 0.292i)15-s + 0.726i·17-s − 0.307i·19-s + (0.748 − 0.529i)21-s + 1.30·23-s + 0.743·25-s + (0.962 − 0.272i)27-s + 0.0362i·29-s − 1.27i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.099908117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.099908117\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (3 + 4.24i)T \) |
| good | 5 | \( 1 + 5.65iT - 125T^{2} \) |
| 7 | \( 1 - 16.9iT - 343T^{2} \) |
| 11 | \( 1 + 30T + 1.33e3T^{2} \) |
| 13 | \( 1 + 72T + 2.19e3T^{2} \) |
| 17 | \( 1 - 50.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 25.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 144T + 1.21e4T^{2} \) |
| 29 | \( 1 - 5.65iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 220. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 72T + 5.06e4T^{2} \) |
| 41 | \( 1 - 305. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 229. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 576T + 1.03e5T^{2} \) |
| 53 | \( 1 - 514. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 414T + 2.05e5T^{2} \) |
| 61 | \( 1 - 504T + 2.26e5T^{2} \) |
| 67 | \( 1 + 789. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 720T + 3.57e5T^{2} \) |
| 73 | \( 1 - 178T + 3.89e5T^{2} \) |
| 79 | \( 1 + 967. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 438T + 5.71e5T^{2} \) |
| 89 | \( 1 + 865. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 650T + 9.12e5T^{2} \) |
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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
−9.790292605824743094080427223395, −8.831152877785787305025275376268, −8.010544043934521718052212966632, −7.22030692245346642423520876066, −6.25018526115930479497867596455, −5.23027544876566832957898144579, −4.82375448064443966960365127689, −2.85115839433250913294710098051, −1.99567842668437017056074980156, −0.52004164675105632566105046199,
0.67223630709930026417278482089, 2.65748778747349539004800100526, 3.59281051872861590536113779537, 4.90369810608827756141043482246, 5.20870754935923210961233770407, 6.82880181912852904291448403733, 7.12976552459949456843422006724, 8.391572935571217812483031024861, 9.559848934418824864060337644834, 10.14157271617555561061806976831
