L(s) = 1 | + (−0.157 − 0.0237i)2-s + (0.607 − 0.414i)3-s + (−1.88 − 0.582i)4-s + (0.234 + 3.13i)5-s + (−0.105 + 0.0508i)6-s + (1.73 − 1.99i)7-s + (0.570 + 0.274i)8-s + (−0.898 + 2.28i)9-s + (0.0373 − 0.498i)10-s + (−1.99 − 5.08i)11-s + (−1.38 + 0.428i)12-s + (0.623 − 0.781i)13-s + (−0.320 + 0.272i)14-s + (1.43 + 1.80i)15-s + (3.17 + 2.16i)16-s + (5.63 + 5.22i)17-s + ⋯ |
L(s) = 1 | + (−0.111 − 0.0167i)2-s + (0.350 − 0.239i)3-s + (−0.943 − 0.291i)4-s + (0.104 + 1.40i)5-s + (−0.0430 + 0.0207i)6-s + (0.656 − 0.754i)7-s + (0.201 + 0.0970i)8-s + (−0.299 + 0.763i)9-s + (0.0118 − 0.157i)10-s + (−0.601 − 1.53i)11-s + (−0.400 + 0.123i)12-s + (0.172 − 0.216i)13-s + (−0.0857 + 0.0729i)14-s + (0.371 + 0.466i)15-s + (0.794 + 0.541i)16-s + (1.36 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25640 + 0.446272i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25640 + 0.446272i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.73 + 1.99i)T \) |
| 13 | \( 1 + (-0.623 + 0.781i)T \) |
good | 2 | \( 1 + (0.157 + 0.0237i)T + (1.91 + 0.589i)T^{2} \) |
| 3 | \( 1 + (-0.607 + 0.414i)T + (1.09 - 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.234 - 3.13i)T + (-4.94 + 0.745i)T^{2} \) |
| 11 | \( 1 + (1.99 + 5.08i)T + (-8.06 + 7.48i)T^{2} \) |
| 17 | \( 1 + (-5.63 - 5.22i)T + (1.27 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.85 - 6.68i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.215 - 0.199i)T + (1.71 - 22.9i)T^{2} \) |
| 29 | \( 1 + (0.882 - 3.86i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.656 + 1.13i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.34 - 1.03i)T + (30.5 - 20.8i)T^{2} \) |
| 41 | \( 1 + (-1.95 - 0.941i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (0.0234 - 0.0113i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-11.2 - 1.69i)T + (44.9 + 13.8i)T^{2} \) |
| 53 | \( 1 + (-7.87 - 2.42i)T + (43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (0.551 - 7.35i)T + (-58.3 - 8.79i)T^{2} \) |
| 61 | \( 1 + (7.39 - 2.28i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (-3.53 + 6.12i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.33 + 5.84i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-2.46 + 0.371i)T + (69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + (6.32 + 10.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.52 - 6.92i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.86 + 4.74i)T + (-65.2 - 60.5i)T^{2} \) |
| 97 | \( 1 + 9.35T + 97T^{2} \) |
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
−10.42932414309717562443529393258, −10.28151463135510476105035522260, −8.725153986196501492871763939519, −7.914966148168593100592529622385, −7.57660179609810026859216134996, −5.93056302985621104408710047975, −5.44946069396338293626875316193, −3.81258467023143849884748169522, −3.10771423361935471604761512280, −1.38110791026349001046141028178,
0.879782606072371666083745707827, 2.64544825485880789588894043861, 4.12442681989263783016010290899, 5.06605162212787214002955774264, 5.34996354053730214963230245785, 7.28204839619476160031213476523, 8.079343685377998117055987445620, 9.043981811901285760554382901993, 9.278042117923106382655363781609, 10.00589132789158104268078782671