L(s) = 1 | + (2.18 − 4.71i)3-s − 14.5i·5-s + 7i·7-s + (−17.4 − 20.6i)9-s − 31.0·11-s − 55.7·13-s + (−68.7 − 31.8i)15-s + 86.4i·17-s − 96.8i·19-s + (32.9 + 15.3i)21-s + 115.·23-s − 87.7·25-s + (−135. + 37.0i)27-s + 49.1i·29-s + 12.5i·31-s + ⋯ |
L(s) = 1 | + (0.420 − 0.907i)3-s − 1.30i·5-s + 0.377i·7-s + (−0.645 − 0.763i)9-s − 0.850·11-s − 1.18·13-s + (−1.18 − 0.549i)15-s + 1.23i·17-s − 1.16i·19-s + (0.342 + 0.159i)21-s + 1.04·23-s − 0.701·25-s + (−0.964 + 0.264i)27-s + 0.314i·29-s + 0.0728i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8652905479\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8652905479\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.18 + 4.71i)T \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 + 14.5iT - 125T^{2} \) |
| 11 | \( 1 + 31.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 55.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 86.4iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 96.8iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 49.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 12.5iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 213. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 165. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 426.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 460. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 686.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 583.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 589. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 766.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 904.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 459. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 119. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 331.T + 9.12e5T^{2} \) |
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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
−10.62297495608501346717467098320, −9.283917423285647442921535570069, −8.709339675023603240243672020173, −7.84971019953567437845796093464, −6.88001144716898422615624887111, −5.54222347150205403428432620229, −4.71028954839488219270176059885, −2.94749932909651930605157095015, −1.70996419128607998155574944977, −0.27293341319712162881370459328,
2.50911486267979513643280238655, 3.23348519471406350390522244835, 4.58106673424733767950532038568, 5.62781066156865040494831897748, 7.14695277747467357174753921196, 7.67658069744261758686704633269, 9.049117468845695478652752812919, 10.05960835111229172960668626100, 10.49853900037268939084902666339, 11.33273375303380445854597305572