L(s) = 1 | + (−2.5 + 4.33i)3-s + (−4.5 − 7.79i)5-s + (−14 + 12.1i)7-s + (0.999 + 1.73i)9-s + (−28.5 + 49.3i)11-s + 70·13-s + 45.0·15-s + (−25.5 + 44.1i)17-s + (2.5 + 4.33i)19-s + (−17.5 − 90.9i)21-s + (−34.5 − 59.7i)23-s + (22 − 38.1i)25-s − 144.·27-s − 114·29-s + (−11.5 + 19.9i)31-s + ⋯ |
L(s) = 1 | + (−0.481 + 0.833i)3-s + (−0.402 − 0.697i)5-s + (−0.755 + 0.654i)7-s + (0.0370 + 0.0641i)9-s + (−0.781 + 1.35i)11-s + 1.49·13-s + 0.774·15-s + (−0.363 + 0.630i)17-s + (0.0301 + 0.0522i)19-s + (−0.181 − 0.944i)21-s + (−0.312 − 0.541i)23-s + (0.175 − 0.304i)25-s − 1.03·27-s − 0.729·29-s + (−0.0666 + 0.115i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (14 - 12.1i)T \) |
good | 3 | \( 1 + (2.5 - 4.33i)T + (-13.5 - 23.3i)T^{2} \) |
| 5 | \( 1 + (4.5 + 7.79i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (28.5 - 49.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 70T + 2.19e3T^{2} \) |
| 17 | \( 1 + (25.5 - 44.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (34.5 + 59.7i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 114T + 2.43e4T^{2} \) |
| 31 | \( 1 + (11.5 - 19.9i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (126.5 + 219. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 42T + 6.89e4T^{2} \) |
| 43 | \( 1 - 124T + 7.95e4T^{2} \) |
| 47 | \( 1 + (100.5 + 174. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (196.5 - 340. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-109.5 + 189. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (354.5 + 614. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-209.5 + 362. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 96T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-156.5 + 271. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (230.5 + 399. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 588T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-508.5 - 880. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.83e3T + 9.12e5T^{2} \) |
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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
−10.45093906725372977515994202723, −9.554064299203757602277690623459, −8.733689343574112062571223348206, −7.78582931463278064391922138549, −6.45364502144832881346747165143, −5.47169597494719786697005321640, −4.57496237718233328623149648036, −3.68703376024002654347866858187, −2.00025111297635820819800713013, 0,
1.13604582103349351145320375145, 3.05980822046880353870539023238, 3.80876302484631115657618471494, 5.63541482430401341330725914353, 6.40059570899197930507171424470, 7.13159965780730776550447186849, 8.002871006466721802766127625533, 9.109733863832044360344825303738, 10.32268203754238261811658589902