| L(s) = 1 | + (−6.42 − 8.83i)2-s + (13.2 + 4.31i)3-s + (−26.9 + 83.0i)4-s + (−22.7 + 51.0i)5-s + (−47.1 − 145. i)6-s + 134. i·7-s + (575. − 186. i)8-s + (−39.0 − 28.3i)9-s + (597. − 126. i)10-s + (−15.5 + 11.2i)11-s + (−716. + 986. i)12-s + (−356. + 491. i)13-s + (1.19e3 − 865. i)14-s + (−522. + 579. i)15-s + (−3.08e3 − 2.24e3i)16-s + (440. − 143. i)17-s + ⋯ |
| L(s) = 1 | + (−1.13 − 1.56i)2-s + (0.851 + 0.276i)3-s + (−0.843 + 2.59i)4-s + (−0.407 + 0.913i)5-s + (−0.534 − 1.64i)6-s + 1.03i·7-s + (3.17 − 1.03i)8-s + (−0.160 − 0.116i)9-s + (1.88 − 0.400i)10-s + (−0.0386 + 0.0280i)11-s + (−1.43 + 1.97i)12-s + (−0.585 + 0.805i)13-s + (1.62 − 1.17i)14-s + (−0.599 + 0.664i)15-s + (−3.01 − 2.18i)16-s + (0.369 − 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.739686 + 0.205944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.739686 + 0.205944i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (22.7 - 51.0i)T \) |
| good | 2 | \( 1 + (6.42 + 8.83i)T + (-9.88 + 30.4i)T^{2} \) |
| 3 | \( 1 + (-13.2 - 4.31i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 - 134. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (15.5 - 11.2i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (356. - 491. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-440. + 143. i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-384. - 1.18e3i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-2.51e3 - 3.46e3i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-783. + 2.41e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (208. + 642. i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-654. + 900. i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (7.61e3 + 5.53e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 4.82e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.48e4 - 4.82e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.10e4 - 3.60e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (1.48e4 + 1.07e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.84e4 - 1.34e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-5.04e4 + 1.63e4i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-1.33e4 + 4.10e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-3.59e4 - 4.94e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.23e4 - 3.80e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-888. + 288. i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-2.04e4 + 1.48e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-1.38e5 - 4.51e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−17.14164598143147113042897532914, −15.40644759544519507920351879465, −13.98341087956908257465462663026, −12.18013255374364084377250897653, −11.38223894348835516897457951898, −9.854839778021677500968453804677, −8.964690782591833271802441911993, −7.67373882318510718602415338957, −3.53284331648816300207843037880, −2.37586224433363709827133460408,
0.70146206529270999915207417119, 5.03693967878723065517229407323, 7.16333169183295317042744808478, 8.097254072152878655852883663998, 9.045008191234101555416817068813, 10.51431850896933504085854095482, 13.21292720845675091728084473723, 14.33418078773446731991284117551, 15.37888269647168394931245604925, 16.67635797691926748045438376312
