L(s) = 1 | + (0.498 + 1.32i)2-s + (−1.50 + 1.31i)4-s + (0.936 + 0.540i)5-s + (0.749 + 2.53i)7-s + (−2.49 − 1.33i)8-s + (−0.249 + 1.50i)10-s + (2.43 + 4.22i)11-s + 0.815·13-s + (−2.98 + 2.25i)14-s + (0.521 − 3.96i)16-s + (1.47 − 0.848i)17-s + (−3.58 − 2.07i)19-s + (−2.12 + 0.421i)20-s + (−4.37 + 5.33i)22-s + (1.75 − 3.04i)23-s + ⋯ |
L(s) = 1 | + (0.352 + 0.935i)2-s + (−0.751 + 0.659i)4-s + (0.418 + 0.241i)5-s + (0.283 + 0.959i)7-s + (−0.881 − 0.471i)8-s + (−0.0787 + 0.477i)10-s + (0.735 + 1.27i)11-s + 0.226·13-s + (−0.797 + 0.602i)14-s + (0.130 − 0.991i)16-s + (0.356 − 0.205i)17-s + (−0.823 − 0.475i)19-s + (−0.474 + 0.0943i)20-s + (−0.933 + 1.13i)22-s + (0.366 − 0.634i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.877 - 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.428114 + 1.67559i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.428114 + 1.67559i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.498 - 1.32i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.749 - 2.53i)T \) |
good | 5 | \( 1 + (-0.936 - 0.540i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.43 - 4.22i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.815T + 13T^{2} \) |
| 17 | \( 1 + (-1.47 + 0.848i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.58 + 2.07i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.75 + 3.04i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9.61iT - 29T^{2} \) |
| 31 | \( 1 + (7.73 - 4.46i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.37 - 5.85i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6.96iT - 41T^{2} \) |
| 43 | \( 1 - 0.510iT - 43T^{2} \) |
| 47 | \( 1 + (-3.40 + 5.90i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.99 + 1.73i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.50 + 4.34i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.85 + 11.8i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.66 + 2.11i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + (1.49 + 2.59i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.41 + 1.97i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + (0.313 + 0.180i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 8.12T + 97T^{2} \) |
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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
−10.59713181069310401890030689490, −9.531229964711973517427841072850, −8.921463042380289625368631289146, −8.130032592071695253643470036983, −6.89821875309804378425265128611, −6.51647255511715623465976440473, −5.31069847034778056678788212833, −4.68553601210973488093450412237, −3.38011797009403046673569210557, −2.00649007790136385273203882105,
0.824484152539137856506283951495, 2.01553227737168553382372671504, 3.64743256014544590068308487107, 4.06521451501897026932111290035, 5.52547209447712180627274275562, 6.06203055484492959788974881517, 7.46254381288659576657983547341, 8.551033305422039984181110718521, 9.289082738746298887780000618779, 10.13354245275593555439560216389