| L(s) = 1 | + (2.73 + 0.732i)2-s + (−6.32 − 10.9i)3-s + (6.92 + 4i)4-s + (24.0 − 24.0i)5-s + (−9.26 − 34.5i)6-s + (−4.96 + 1.33i)7-s + (15.9 + 16i)8-s + (−39.6 + 68.6i)9-s + (83.3 − 48.1i)10-s + (−57.5 + 214. i)11-s − 101. i·12-s + (109. − 128. i)13-s − 14.5·14-s + (−415. − 111. i)15-s + (31.9 + 55.4i)16-s + (32.5 + 18.7i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (−0.703 − 1.21i)3-s + (0.433 + 0.250i)4-s + (0.962 − 0.962i)5-s + (−0.257 − 0.960i)6-s + (−0.101 + 0.0271i)7-s + (0.249 + 0.250i)8-s + (−0.489 + 0.847i)9-s + (0.833 − 0.481i)10-s + (−0.475 + 1.77i)11-s − 0.703i·12-s + (0.649 − 0.760i)13-s − 0.0741·14-s + (−1.84 − 0.495i)15-s + (0.124 + 0.216i)16-s + (0.112 + 0.0650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.544 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.52251 - 0.826327i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52251 - 0.826327i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.73 - 0.732i)T \) |
| 13 | \( 1 + (-109. + 128. i)T \) |
| good | 3 | \( 1 + (6.32 + 10.9i)T + (-40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-24.0 + 24.0i)T - 625iT^{2} \) |
| 7 | \( 1 + (4.96 - 1.33i)T + (2.07e3 - 1.20e3i)T^{2} \) |
| 11 | \( 1 + (57.5 - 214. i)T + (-1.26e4 - 7.32e3i)T^{2} \) |
| 17 | \( 1 + (-32.5 - 18.7i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-120. - 448. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (-299. + 172. i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (430. + 746. i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-454. + 454. i)T - 9.23e5iT^{2} \) |
| 37 | \( 1 + (590. - 2.20e3i)T + (-1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + (-632. - 169. i)T + (2.44e6 + 1.41e6i)T^{2} \) |
| 43 | \( 1 + (1.01e3 + 585. i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (2.74e3 + 2.74e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 - 1.14e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (4.97e3 - 1.33e3i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (285. - 493. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.95e3 - 792. i)T + (1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 + (820. + 3.06e3i)T + (-2.20e7 + 1.27e7i)T^{2} \) |
| 73 | \( 1 + (-2.09e3 - 2.09e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 3.58e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-5.09e3 + 5.09e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (-2.65e3 + 9.90e3i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-2.38e3 - 8.88e3i)T + (-7.66e7 + 4.42e7i)T^{2} \) |
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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
−16.78880236747012243530663962854, −15.18242233544546890711705113037, −13.42939209093771751987616104507, −12.84617058958767622197718670996, −11.99320812437435034130993024956, −9.992364125770145573751312407625, −7.84246524927249484231353454575, −6.31510679171961504843262698072, −5.13167804865329644740065733435, −1.65896073214218795902142268379,
3.27212779336440379116597521495, 5.27788892700977412759158376649, 6.42303068141771569396884377272, 9.352172290860617071316002554687, 10.84827424931964137898582761130, 11.14147199715221880578601365945, 13.41265370237228742133899997273, 14.31634944048114834934723365431, 15.74765115806918779980852502834, 16.52714503994036696122979107390
