| L(s) = 1 | + (5.71 − 1.85i)2-s + (−10.7 − 14.8i)3-s + (3.32 − 2.41i)4-s + (37.8 − 41.1i)5-s + (−89.2 − 64.8i)6-s − 177. i·7-s + (−98.5 + 135. i)8-s + (−28.9 + 89.2i)9-s + (140. − 305. i)10-s + (174. + 536. i)11-s + (−71.8 − 23.3i)12-s + (578. + 188. i)13-s + (−328. − 1.01e3i)14-s + (−1.01e3 − 118. i)15-s + (−351. + 1.08e3i)16-s + (950. − 1.30e3i)17-s + ⋯ |
| L(s) = 1 | + (1.01 − 0.328i)2-s + (−0.692 − 0.952i)3-s + (0.104 − 0.0755i)4-s + (0.677 − 0.735i)5-s + (−1.01 − 0.735i)6-s − 1.36i·7-s + (−0.544 + 0.748i)8-s + (−0.119 + 0.367i)9-s + (0.443 − 0.965i)10-s + (0.434 + 1.33i)11-s + (−0.143 − 0.0467i)12-s + (0.949 + 0.308i)13-s + (−0.448 − 1.38i)14-s + (−1.16 − 0.136i)15-s + (−0.343 + 1.05i)16-s + (0.797 − 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.168 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.26364 - 1.49792i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26364 - 1.49792i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-37.8 + 41.1i)T \) |
| good | 2 | \( 1 + (-5.71 + 1.85i)T + (25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (10.7 + 14.8i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 + 177. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-174. - 536. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-578. - 188. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-950. + 1.30e3i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.26e3 - 921. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (1.77e3 - 576. i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (450. - 326. i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (4.54e3 + 3.30e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-7.49e3 - 2.43e3i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (144. - 444. i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 5.24e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.28e4 - 1.77e4i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-6.14e3 - 8.46e3i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-3.19e3 + 9.82e3i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-1.09e3 - 3.35e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (1.01e4 - 1.39e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (3.18e4 - 2.31e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (6.93e4 - 2.25e4i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (2.98e4 - 2.16e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-1.05e4 + 1.45e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (-2.68e4 - 8.25e4i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (6.88e4 + 9.47e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
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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
−16.49139680175616534549024951533, −14.25651214768108861373071940186, −13.44629298544413607329842110604, −12.54599975838956440030948379636, −11.58555235179376159567089578584, −9.651496996617440347060530185451, −7.42050637809059335108815010369, −5.82429626805141716760159840347, −4.22981008743864673048740789497, −1.29639921694876882136815720655,
3.49301965078223867741540585249, 5.60463006301962853000511171157, 5.96813438960283356421980462357, 9.030104257862499302842838033848, 10.45789039698263352957878131224, 11.73394625313864265463445115929, 13.34453033317282617982868177348, 14.52013879825626476315036333063, 15.49747589201732971959968470180, 16.42874076298370639488526195123
