| L(s) = 1 | + (−0.312 − 1.97i)2-s + (−4.02 − 2.04i)3-s + (0.00704 − 0.00229i)4-s + (4.93 + 0.810i)5-s + (−2.78 + 8.57i)6-s + (3.91 + 3.91i)7-s + (−3.63 − 7.13i)8-s + (6.67 + 9.19i)9-s + (0.0571 − 9.99i)10-s + (−2.73 − 1.98i)11-s + (−0.0330 − 0.00523i)12-s + (−3.13 + 19.8i)13-s + (6.49 − 8.94i)14-s + (−18.1 − 13.3i)15-s + (−12.9 + 9.38i)16-s + (17.7 − 9.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.156 − 0.986i)2-s + (−1.34 − 0.682i)3-s + (0.00176 − 0.000572i)4-s + (0.986 + 0.162i)5-s + (−0.464 + 1.42i)6-s + (0.558 + 0.558i)7-s + (−0.454 − 0.891i)8-s + (0.742 + 1.02i)9-s + (0.00571 − 0.999i)10-s + (−0.248 − 0.180i)11-s + (−0.00275 − 0.000435i)12-s + (−0.241 + 1.52i)13-s + (0.464 − 0.638i)14-s + (−1.21 − 0.891i)15-s + (−0.807 + 0.586i)16-s + (1.04 − 0.530i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.494270 - 0.567141i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.494270 - 0.567141i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.93 - 0.810i)T \) |
| good | 2 | \( 1 + (0.312 + 1.97i)T + (-3.80 + 1.23i)T^{2} \) |
| 3 | \( 1 + (4.02 + 2.04i)T + (5.29 + 7.28i)T^{2} \) |
| 7 | \( 1 + (-3.91 - 3.91i)T + 49iT^{2} \) |
| 11 | \( 1 + (2.73 + 1.98i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (3.13 - 19.8i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-17.7 + 9.02i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (5.87 + 1.91i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (13.6 - 2.16i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (29.3 - 9.52i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (0.774 - 2.38i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (33.6 + 5.32i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-29.0 + 21.0i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-23.3 + 23.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (19.2 - 37.6i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (73.4 + 37.4i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-1.53 - 2.11i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-21.8 - 15.8i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-67.8 + 34.5i)T + (2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-22.6 - 69.7i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-51.6 + 8.18i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (37.4 - 12.1i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (22.8 + 44.7i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (41.7 - 57.4i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (1.26 - 2.48i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−17.39234459512347619056049811187, −16.24458537924087836675632854497, −14.25268908180441249693078254991, −12.67466200217122299358718064995, −11.76121727974677572219941437083, −10.88964050137838133853927003188, −9.485546509614340535996700693244, −6.83345711223090446186017418201, −5.54141734882873054540372143661, −1.84533285430714077018680931209,
5.19594970885838571033013547476, 6.06089592288810378399494501234, 7.86519325979508714283963360305, 10.01333861795390442885845117091, 10.98159131166635095353004679255, 12.57144314245464428903341573801, 14.41016848360566460328866560127, 15.58152926586773706260758626119, 16.77901279097029436469153183864, 17.29295625807971537709185432295
