L(s) = 1 | + (−0.662 − 1.24i)2-s + (0.980 − 0.195i)3-s + (−1.12 + 1.65i)4-s + (1.62 + 2.43i)5-s + (−0.893 − 1.09i)6-s + (0.294 + 0.121i)7-s + (2.81 + 0.303i)8-s + (0.923 − 0.382i)9-s + (1.96 − 3.64i)10-s + (−1.07 + 5.42i)11-s + (−0.776 + 1.84i)12-s + (2.67 − 4.00i)13-s + (−0.0427 − 0.448i)14-s + (2.06 + 2.06i)15-s + (−1.48 − 3.71i)16-s + (0.394 − 0.394i)17-s + ⋯ |
L(s) = 1 | + (−0.468 − 0.883i)2-s + (0.566 − 0.112i)3-s + (−0.560 + 0.828i)4-s + (0.726 + 1.08i)5-s + (−0.364 − 0.447i)6-s + (0.111 + 0.0460i)7-s + (0.994 + 0.107i)8-s + (0.307 − 0.127i)9-s + (0.620 − 1.15i)10-s + (−0.325 + 1.63i)11-s + (−0.224 + 0.532i)12-s + (0.742 − 1.11i)13-s + (−0.0114 − 0.119i)14-s + (0.534 + 0.534i)15-s + (−0.371 − 0.928i)16-s + (0.0956 − 0.0956i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17854 - 0.185972i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17854 - 0.185972i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.662 + 1.24i)T \) |
| 3 | \( 1 + (-0.980 + 0.195i)T \) |
good | 5 | \( 1 + (-1.62 - 2.43i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.294 - 0.121i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (1.07 - 5.42i)T + (-10.1 - 4.20i)T^{2} \) |
| 13 | \( 1 + (-2.67 + 4.00i)T + (-4.97 - 12.0i)T^{2} \) |
| 17 | \( 1 + (-0.394 + 0.394i)T - 17iT^{2} \) |
| 19 | \( 1 + (2.13 + 1.42i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (3.37 + 8.14i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.411 - 2.06i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + 1.31iT - 31T^{2} \) |
| 37 | \( 1 + (-1.47 + 0.985i)T + (14.1 - 34.1i)T^{2} \) |
| 41 | \( 1 + (-4.39 - 10.6i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (2.25 + 0.448i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (5.23 - 5.23i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.88 + 9.48i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (0.373 + 0.559i)T + (-22.5 + 54.5i)T^{2} \) |
| 61 | \( 1 + (-0.868 + 0.172i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 + (-10.3 + 2.04i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (12.4 + 5.14i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (14.5 - 6.01i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (2.55 + 2.55i)T + 79iT^{2} \) |
| 83 | \( 1 + (-0.585 - 0.391i)T + (31.7 + 76.6i)T^{2} \) |
| 89 | \( 1 + (-3.72 + 8.98i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 0.565iT - 97T^{2} \) |
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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
−12.69495690650349473052460996406, −11.28793307679865917144465427419, −10.20551808130957762823837857789, −9.973384078191791899665211146471, −8.574187865129933262855337634128, −7.61289739253721692827886320625, −6.46362190433189498440202629047, −4.58375552892895446224918453452, −3.01407913868063894585617439972, −2.06731955118537593672009647082,
1.50169079904368332106909589720, 3.98192082552419725095957937372, 5.42032134508362379393035155445, 6.19883870616023039094781469260, 7.77016604930054822889678573253, 8.723598451320258031736043241410, 9.142831042440713037950059770748, 10.26401879317275472515813093156, 11.46675312856691627728944358607, 13.11190636734431762939771475807