L(s) = 1 | + (1.5 + 2.59i)3-s + (−4.91 + 8.50i)5-s + (−16.3 + 8.74i)7-s + (−4.5 + 7.79i)9-s + (7.08 + 12.2i)11-s − 26.1·13-s − 29.4·15-s + (39.2 + 68.0i)17-s + (−36.5 + 63.3i)19-s + (−47.2 − 29.2i)21-s + (48 − 83.1i)23-s + (14.2 + 24.6i)25-s − 27·27-s + 173.·29-s + (−33.6 − 58.2i)31-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.439 + 0.761i)5-s + (−0.881 + 0.472i)7-s + (−0.166 + 0.288i)9-s + (0.194 + 0.336i)11-s − 0.557·13-s − 0.507·15-s + (0.560 + 0.971i)17-s + (−0.441 + 0.765i)19-s + (−0.490 − 0.304i)21-s + (0.435 − 0.753i)23-s + (0.113 + 0.197i)25-s − 0.192·27-s + 1.10·29-s + (−0.194 − 0.337i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.557432 + 1.00177i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.557432 + 1.00177i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 2.59i)T \) |
| 7 | \( 1 + (16.3 - 8.74i)T \) |
good | 5 | \( 1 + (4.91 - 8.50i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-7.08 - 12.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 26.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-39.2 - 68.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (36.5 - 63.3i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-48 + 83.1i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 + (33.6 + 58.2i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-150. + 261. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 472.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 463.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-45.5 + 78.9i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-81.6 - 141. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-300. - 520. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (285. - 495. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-269. - 467. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 1.06e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-221. - 383. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-22.8 + 39.5i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 686.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (330. - 571. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 658.T + 9.12e5T^{2} \) |
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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
−14.59353375711356349416584337107, −12.96641778209182791853576527243, −12.04753149751472493880809761521, −10.66563070725423610220240504987, −9.849679006616358412002857782343, −8.601523118034197292544683707033, −7.23113562034042865248576463590, −5.93316820101530037023887603126, −4.07236782090922010491321613640, −2.75070442573841578542245323906,
0.68426468633149209831045886963, 3.08858726264502574249966638880, 4.76956288459268093062528512620, 6.52674765005095420558009850269, 7.64725174898921561250186042742, 8.878496162589334890971248536935, 9.903057498267390880780454034494, 11.48129422008524655688178624662, 12.49066990568491455426125559000, 13.30177007856094058250977685806