| L(s) = 1 | + (−21.2 + 3.75i)2-s + (241. + 288. i)3-s + (−522. + 190. i)4-s + (−1.31e3 − 477. i)5-s + (−6.23e3 − 5.22e3i)6-s + (4.89e3 − 8.48e3i)7-s + (2.95e4 − 1.70e4i)8-s + (−1.43e4 + 8.11e4i)9-s + (2.97e4 + 5.24e3i)10-s + (2.14e4 + 3.71e4i)11-s + (−1.81e5 − 1.04e5i)12-s + (−6.26e3 + 7.46e3i)13-s + (−7.24e4 + 1.99e5i)14-s + (−1.79e5 − 4.93e5i)15-s + (−1.29e5 + 1.09e5i)16-s + (−3.16e5 − 1.79e6i)17-s + ⋯ |
| L(s) = 1 | + (−0.665 + 0.117i)2-s + (0.994 + 1.18i)3-s + (−0.510 + 0.185i)4-s + (−0.420 − 0.152i)5-s + (−0.801 − 0.672i)6-s + (0.291 − 0.504i)7-s + (0.903 − 0.521i)8-s + (−0.242 + 1.37i)9-s + (0.297 + 0.0524i)10-s + (0.133 + 0.230i)11-s + (−0.728 − 0.420i)12-s + (−0.0168 + 0.0200i)13-s + (−0.134 + 0.370i)14-s + (−0.236 − 0.650i)15-s + (−0.123 + 0.103i)16-s + (−0.223 − 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.03273 + 1.28423i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03273 + 1.28423i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.31e3 + 477. i)T \) |
| 19 | \( 1 + (8.03e5 - 2.34e6i)T \) |
| good | 2 | \( 1 + (21.2 - 3.75i)T + (962. - 350. i)T^{2} \) |
| 3 | \( 1 + (-241. - 288. i)T + (-1.02e4 + 5.81e4i)T^{2} \) |
| 7 | \( 1 + (-4.89e3 + 8.48e3i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (-2.14e4 - 3.71e4i)T + (-1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (6.26e3 - 7.46e3i)T + (-2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (3.16e5 + 1.79e6i)T + (-1.89e12 + 6.89e11i)T^{2} \) |
| 23 | \( 1 + (-4.54e6 + 1.65e6i)T + (3.17e13 - 2.66e13i)T^{2} \) |
| 29 | \( 1 + (-2.73e7 - 4.82e6i)T + (3.95e14 + 1.43e14i)T^{2} \) |
| 31 | \( 1 + (-3.75e7 - 2.16e7i)T + (4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + 2.08e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + (1.44e7 + 1.72e7i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (-6.80e7 - 2.47e7i)T + (1.65e16 + 1.38e16i)T^{2} \) |
| 47 | \( 1 + (-4.15e7 + 2.35e8i)T + (-4.94e16 - 1.79e16i)T^{2} \) |
| 53 | \( 1 + (-1.02e8 - 2.82e8i)T + (-1.33e17 + 1.12e17i)T^{2} \) |
| 59 | \( 1 + (4.38e8 - 7.74e7i)T + (4.80e17 - 1.74e17i)T^{2} \) |
| 61 | \( 1 + (3.78e8 - 1.37e8i)T + (5.46e17 - 4.58e17i)T^{2} \) |
| 67 | \( 1 + (1.45e9 + 2.55e8i)T + (1.71e18 + 6.23e17i)T^{2} \) |
| 71 | \( 1 + (3.65e8 - 1.00e9i)T + (-2.49e18 - 2.09e18i)T^{2} \) |
| 73 | \( 1 + (1.52e9 - 1.27e9i)T + (7.46e17 - 4.23e18i)T^{2} \) |
| 79 | \( 1 + (-1.09e9 - 1.30e9i)T + (-1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (2.86e9 - 4.96e9i)T + (-7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 + (2.51e9 - 2.99e9i)T + (-5.41e18 - 3.07e19i)T^{2} \) |
| 97 | \( 1 + (-5.73e9 + 1.01e9i)T + (6.92e19 - 2.52e19i)T^{2} \) |
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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
−12.23519781009520664064488353054, −10.66106595195918341536961271836, −9.919709726405362555582093114767, −8.946965916338636953160340716594, −8.277722975494140656602249858519, −7.19151015255429518761068906917, −4.78209546916894274360934212936, −4.17406846419215054006123423494, −2.94320871421762043920627187630, −0.953318133313329595945561589724,
0.61752090485601438970271378767, 1.68265719659594701828638497762, 2.85336454821827267819798866178, 4.51894665052721184827350055269, 6.32058236080419115987191334525, 7.62943145619527855230635312587, 8.435698982552553155165207045044, 9.006602258349660154801857209247, 10.46081110665556930347645318011, 11.72325706279870751131658909380
