| L(s) = 1 | + (−9.71 + 3.15i)2-s + (−6.12 − 8.42i)3-s + (58.4 − 42.4i)4-s + (−38.3 − 40.6i)5-s + (86.0 + 62.5i)6-s + 211. i·7-s + (−241. + 332. i)8-s + (41.5 − 127. i)9-s + (500. + 274. i)10-s + (137. + 424. i)11-s + (−716. − 232. i)12-s + (418. + 136. i)13-s + (−668. − 2.05e3i)14-s + (−108. + 572. i)15-s + (583. − 1.79e3i)16-s + (142. − 196. i)17-s + ⋯ |
| L(s) = 1 | + (−1.71 + 0.557i)2-s + (−0.392 − 0.540i)3-s + (1.82 − 1.32i)4-s + (−0.685 − 0.727i)5-s + (0.975 + 0.709i)6-s + 1.63i·7-s + (−1.33 + 1.83i)8-s + (0.171 − 0.526i)9-s + (1.58 + 0.866i)10-s + (0.343 + 1.05i)11-s + (−1.43 − 0.466i)12-s + (0.686 + 0.223i)13-s + (−0.911 − 2.80i)14-s + (−0.124 + 0.656i)15-s + (0.569 − 1.75i)16-s + (0.119 − 0.164i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.368939 + 0.269032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.368939 + 0.269032i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (38.3 + 40.6i)T \) |
| good | 2 | \( 1 + (9.71 - 3.15i)T + (25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (6.12 + 8.42i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 211. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-137. - 424. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (-418. - 136. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-142. + 196. i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.82e3 - 1.32e3i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-89.2 + 28.9i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (1.04e3 - 761. i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.26e3 - 919. i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (-107. - 35.0i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (3.70e3 - 1.14e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 + 1.21e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-5.59e3 - 7.70e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-1.81e4 - 2.49e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (1.30e4 - 4.02e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-2.89e3 - 8.91e3i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (-4.36e3 + 6.00e3i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (-4.88e4 + 3.55e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-2.01e3 + 656. i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (5.59e4 - 4.06e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (9.94e3 - 1.36e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (3.74e4 + 1.15e5i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (5.38e4 + 7.40e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
| 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
−16.96634364555981829237299499877, −15.80311549917250826670799702603, −15.11481365360589351927239523374, −12.32267199015475789783177214671, −11.65751837691089093828656181006, −9.570507516393800630143988867884, −8.688081328606594559332976165110, −7.34376557269538768919278507877, −5.85120100642107314960214474250, −1.37496393318725469279174275021,
0.64425656919755241137997405016, 3.57392027587984348208475931840, 7.03386194415418462233026619863, 8.125007926177083562580195672673, 9.936183645368338377496934622749, 10.91844789515762170049782882301, 11.34857710786508850687441527984, 13.72443413499626492901221127376, 15.83926852113011446560762533100, 16.52939307180549268662185113519
