L(s) = 1 | + (−1.36 + 0.386i)2-s + (0.707 − 0.707i)3-s + (1.70 − 1.05i)4-s + (−1.98 + 1.03i)5-s + (−0.688 + 1.23i)6-s − 3.91·7-s + (−1.90 + 2.08i)8-s − 1.00i·9-s + (2.30 − 2.16i)10-s + (−2.93 + 2.93i)11-s + (0.459 − 1.94i)12-s + (−0.732 + 0.732i)13-s + (5.33 − 1.51i)14-s + (−0.674 + 2.13i)15-s + (1.78 − 3.57i)16-s + 2.89i·17-s + ⋯ |
L(s) = 1 | + (−0.961 + 0.273i)2-s + (0.408 − 0.408i)3-s + (0.850 − 0.525i)4-s + (−0.887 + 0.461i)5-s + (−0.281 + 0.504i)6-s − 1.48·7-s + (−0.674 + 0.738i)8-s − 0.333i·9-s + (0.727 − 0.686i)10-s + (−0.884 + 0.884i)11-s + (0.132 − 0.561i)12-s + (−0.203 + 0.203i)13-s + (1.42 − 0.404i)14-s + (−0.174 + 0.550i)15-s + (0.447 − 0.894i)16-s + 0.701i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0347329 + 0.182018i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0347329 + 0.182018i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 - 0.386i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (1.98 - 1.03i)T \) |
good | 7 | \( 1 + 3.91T + 7T^{2} \) |
| 11 | \( 1 + (2.93 - 2.93i)T - 11iT^{2} \) |
| 13 | \( 1 + (0.732 - 0.732i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.89iT - 17T^{2} \) |
| 19 | \( 1 + (-1.67 - 1.67i)T + 19iT^{2} \) |
| 23 | \( 1 + 1.73T + 23T^{2} \) |
| 29 | \( 1 + (4.99 + 4.99i)T + 29iT^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 + (-6.41 - 6.41i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.00577iT - 41T^{2} \) |
| 43 | \( 1 + (2.23 + 2.23i)T + 43iT^{2} \) |
| 47 | \( 1 + 11.6iT - 47T^{2} \) |
| 53 | \( 1 + (-5.55 - 5.55i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.83 + 3.83i)T - 59iT^{2} \) |
| 61 | \( 1 + (-9.30 - 9.30i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.85 - 3.85i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.15iT - 71T^{2} \) |
| 73 | \( 1 + 7.98T + 73T^{2} \) |
| 79 | \( 1 - 0.843T + 79T^{2} \) |
| 83 | \( 1 + (5.20 - 5.20i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.40iT - 89T^{2} \) |
| 97 | \( 1 + 2.24iT - 97T^{2} \) |
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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
−12.46729367445092420720111037060, −11.53879908261145904970603923472, −10.28533244540065680087018928373, −9.728926926505562919066657385995, −8.550147206217887043633885583661, −7.50640479212057468819701814543, −6.99492864091334947758360239943, −5.82746414636844987160710794559, −3.68551566358648944133981751977, −2.38611960591566056572757190053,
0.17779941530162031926979124300, 2.87855356431618703637531534917, 3.69364430471320310713302760047, 5.57101046205488799015121828469, 7.10875433680665381091824467100, 7.890114994299573337157803115613, 9.025614276879992661534536723333, 9.528136080784193357151040197722, 10.69215615457489155075092886034, 11.42858734976111189381039273728