| L(s) = 1 | + (0.638 − 1.26i)2-s + (0.278 − 1.70i)3-s + (−1.18 − 1.61i)4-s + (2.36 − 0.206i)5-s + (−1.97 − 1.44i)6-s + (1.05 − 2.89i)7-s + (−2.79 + 0.463i)8-s + (−2.84 − 0.951i)9-s + (1.24 − 3.11i)10-s + (−0.494 + 5.65i)11-s + (−3.08 + 1.57i)12-s + (2.53 − 1.77i)13-s + (−2.97 − 3.17i)14-s + (0.303 − 4.09i)15-s + (−1.19 + 3.81i)16-s + (2.45 − 4.25i)17-s + ⋯ |
| L(s) = 1 | + (0.451 − 0.892i)2-s + (0.160 − 0.987i)3-s + (−0.591 − 0.806i)4-s + (1.05 − 0.0923i)5-s + (−0.807 − 0.589i)6-s + (0.398 − 1.09i)7-s + (−0.986 + 0.163i)8-s + (−0.948 − 0.317i)9-s + (0.394 − 0.983i)10-s + (−0.149 + 1.70i)11-s + (−0.890 + 0.454i)12-s + (0.704 − 0.493i)13-s + (−0.795 − 0.849i)14-s + (0.0784 − 1.05i)15-s + (−0.299 + 0.954i)16-s + (0.595 − 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.528412 - 1.92655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.528412 - 1.92655i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.638 + 1.26i)T \) |
| 3 | \( 1 + (-0.278 + 1.70i)T \) |
| good | 5 | \( 1 + (-2.36 + 0.206i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-1.05 + 2.89i)T + (-5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (0.494 - 5.65i)T + (-10.8 - 1.91i)T^{2} \) |
| 13 | \( 1 + (-2.53 + 1.77i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-2.45 + 4.25i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.14 - 4.25i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-1.86 - 5.11i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-4.02 + 5.75i)T + (-9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (8.66 - 3.15i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.57 - 5.89i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.25 + 0.396i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.772 + 8.83i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-3.18 - 1.15i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (3.25 - 3.25i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.81 - 0.246i)T + (58.1 - 10.2i)T^{2} \) |
| 61 | \( 1 + (2.03 + 4.36i)T + (-39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (-9.01 + 6.30i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (-8.19 - 4.73i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.68 - 2.13i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.0453 + 0.256i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-6.79 + 9.69i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-7.23 + 4.17i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.03 - 5.90i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−10.78366086654925698856369350922, −9.990244889910211144812496289236, −9.306758787448740858828508191294, −7.922847071790788094813716667225, −7.04031081936537600285429084469, −5.86704171538954708320417686585, −4.96573495245169406213444305225, −3.57131753806183780301602388912, −2.11721603713834484539086489953, −1.25757191634578589533670311733,
2.61140562366281717800514184382, 3.73084931590018324567795078480, 5.10011653862453186342740563660, 5.78976627644889761677383264846, 6.36500765465519165225562056010, 8.205200657147976746228554800362, 8.795826630116790040741560529159, 9.309358792632846795686012502122, 10.69624138954773505217196012371, 11.34471798508745539478533903547
