L(s) = 1 | + (−5.09 + 1.04i)3-s − 1.90·5-s + (3.04 − 18.2i)7-s + (24.8 − 10.6i)9-s + 38.8i·11-s + (8.40 − 4.85i)13-s + (9.67 − 1.97i)15-s + (−22.9 − 39.6i)17-s + (19.7 + 11.4i)19-s + (3.49 + 96.1i)21-s + 156. i·23-s − 121.·25-s + (−115. + 79.8i)27-s + (−187. − 108. i)29-s + (−174. − 100. i)31-s + ⋯ |
L(s) = 1 | + (−0.979 + 0.200i)3-s − 0.169·5-s + (0.164 − 0.986i)7-s + (0.919 − 0.392i)9-s + 1.06i·11-s + (0.179 − 0.103i)13-s + (0.166 − 0.0340i)15-s + (−0.326 − 0.565i)17-s + (0.239 + 0.138i)19-s + (0.0363 + 0.999i)21-s + 1.42i·23-s − 0.971·25-s + (−0.822 + 0.568i)27-s + (−1.19 − 0.692i)29-s + (−1.00 − 0.582i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.3259889306\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3259889306\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.09 - 1.04i)T \) |
| 7 | \( 1 + (-3.04 + 18.2i)T \) |
good | 5 | \( 1 + 1.90T + 125T^{2} \) |
| 11 | \( 1 - 38.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-8.40 + 4.85i)T + (1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 + (22.9 + 39.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-19.7 - 11.4i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 - 156. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (187. + 108. i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (174. + 100. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (146. - 253. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-119. - 206. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (158. - 275. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-76.4 - 132. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-26.3 + 15.1i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (233. - 404. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-439. + 253. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (315. - 546. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 413. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-276. + 159. i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (502. + 870. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (415. - 719. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-572. + 990. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.55e3 + 899. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−11.66467683101514478652790060234, −11.25162644967513866219923272508, −10.04069586916696362932716325952, −9.524287851675984348590065330069, −7.68855422477658082892877983902, −7.12404952936082034352816736621, −5.84290181039976395790543007087, −4.70291587906393080517862291958, −3.77442631635276740063910734959, −1.51193794310523599476447807527,
0.14832500652789629879604350140, 1.96057731666962045645807993942, 3.78073257555201731648355776754, 5.28039759686765783629205320040, 5.94802715687878407922892867707, 7.04861775627951044555505174094, 8.329817741265384885396732653097, 9.190954163182104697337177168698, 10.64136318558419203599740998732, 11.17248314741275452477256520773