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 | \( 1 + (-2.38e3 + 8.88e3i)T + (-7.66e7 - 4.42e7i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.52714503994036696122979107390, −15.74765115806918779980852502834, −14.31634944048114834934723365431, −13.41265370237228742133899997273, −11.14147199715221880578601365945, −10.84827424931964137898582761130, −9.352172290860617071316002554687, −6.42303068141771569396884377272, −5.27788892700977412759158376649, −3.27212779336440379116597521495,
1.65896073214218795902142268379, 5.13167804865329644740065733435, 6.31510679171961504843262698072, 7.84246524927249484231353454575, 9.992364125770145573751312407625, 11.99320812437435034130993024956, 12.84617058958767622197718670996, 13.42939209093771751987616104507, 15.18242233544546890711705113037, 16.78880236747012243530663962854