| L(s) = 1 | + (0.958 − 1.31i)2-s + (−8.25 + 2.68i)3-s + (9.06 + 27.9i)4-s + (−55.8 + 3.21i)5-s + (−4.37 + 13.4i)6-s + 127. i·7-s + (95.0 + 30.8i)8-s + (−135. + 98.5i)9-s + (−49.2 + 76.6i)10-s + (416. + 302. i)11-s + (−149. − 206. i)12-s + (−641. − 883. i)13-s + (167. + 121. i)14-s + (452. − 176. i)15-s + (−627. + 456. i)16-s + (1.00e3 + 325. i)17-s + ⋯ |
| L(s) = 1 | + (0.169 − 0.233i)2-s + (−0.529 + 0.172i)3-s + (0.283 + 0.872i)4-s + (−0.998 + 0.0575i)5-s + (−0.0495 + 0.152i)6-s + 0.981i·7-s + (0.525 + 0.170i)8-s + (−0.558 + 0.405i)9-s + (−0.155 + 0.242i)10-s + (1.03 + 0.753i)11-s + (−0.300 − 0.413i)12-s + (−1.05 − 1.45i)13-s + (0.228 + 0.166i)14-s + (0.518 − 0.202i)15-s + (−0.613 + 0.445i)16-s + (0.840 + 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.587993 + 0.800404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.587993 + 0.800404i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (55.8 - 3.21i)T \) |
| good | 2 | \( 1 + (-0.958 + 1.31i)T + (-9.88 - 30.4i)T^{2} \) |
| 3 | \( 1 + (8.25 - 2.68i)T + (196. - 142. i)T^{2} \) |
| 7 | \( 1 - 127. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-416. - 302. i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (641. + 883. i)T + (-1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.00e3 - 325. i)T + (1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-321. + 989. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (18.0 - 24.8i)T + (-1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-2.24e3 - 6.91e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (3.01e3 - 9.27e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-2.79e3 - 3.85e3i)T + (-2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.72e3 + 1.25e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.15e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.28e4 + 4.19e3i)T + (1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.52e4 - 4.95e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.21e3 + 1.61e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-3.50e3 - 2.54e3i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (2.25e4 + 7.32e3i)T + (1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (3.09e3 + 9.53e3i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.02e4 - 1.40e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (4.52e3 + 1.39e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-6.26e4 - 2.03e4i)T + (3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (1.82e4 + 1.32e4i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (9.01e4 - 2.92e4i)T + (6.94e9 - 5.04e9i)T^{2} \) |
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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
−16.89648058556446990014716727451, −15.75604853109030888065120988348, −14.63216985472374641653245636291, −12.32131751861080042536421844418, −12.14728602666786333158435913247, −10.72800148886726501659006993805, −8.602585356721272679407993407867, −7.25880696340782769062960372488, −5.04208192040254439584651963409, −3.05164966825618322618971688822,
0.69967428894607806787734695007, 4.21607382015682598661925850876, 6.16495308899724524242102869666, 7.42643567612333294084614094007, 9.557664472508133433611731661025, 11.25062874455081669048171475280, 11.89818737542885932802912595105, 14.04642885662701039863884076433, 14.73865994992763919197226852695, 16.43121850850026151492601422708
