| L(s) = 1 | − i·3-s + i·5-s − 7-s − 9-s − 2.61i·11-s − 0.215i·13-s + 15-s + 1.53·17-s + 4.57i·19-s + i·21-s + 2.46·23-s − 25-s + i·27-s + 2.86i·29-s + 2.30·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447i·5-s − 0.377·7-s − 0.333·9-s − 0.787i·11-s − 0.0597i·13-s + 0.258·15-s + 0.371·17-s + 1.04i·19-s + 0.218i·21-s + 0.514·23-s − 0.200·25-s + 0.192i·27-s + 0.532i·29-s + 0.414·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.679726342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.679726342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2.61iT - 11T^{2} \) |
| 13 | \( 1 + 0.215iT - 13T^{2} \) |
| 17 | \( 1 - 1.53T + 17T^{2} \) |
| 19 | \( 1 - 4.57iT - 19T^{2} \) |
| 23 | \( 1 - 2.46T + 23T^{2} \) |
| 29 | \( 1 - 2.86iT - 29T^{2} \) |
| 31 | \( 1 - 2.30T + 31T^{2} \) |
| 37 | \( 1 - 1.53iT - 37T^{2} \) |
| 41 | \( 1 - 8.05T + 41T^{2} \) |
| 43 | \( 1 - 1.69iT - 43T^{2} \) |
| 47 | \( 1 + 3.53T + 47T^{2} \) |
| 53 | \( 1 + 3.55iT - 53T^{2} \) |
| 59 | \( 1 - 1.06iT - 59T^{2} \) |
| 61 | \( 1 + 11.9iT - 61T^{2} \) |
| 67 | \( 1 + 6.25iT - 67T^{2} \) |
| 71 | \( 1 - 2.18T + 71T^{2} \) |
| 73 | \( 1 + 0.849T + 73T^{2} \) |
| 79 | \( 1 - 4.72T + 79T^{2} \) |
| 83 | \( 1 - 6.49iT - 83T^{2} \) |
| 89 | \( 1 - 9.55T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.400594131352579076200969488462, −7.84773121668448547668214183838, −7.07006797418073860598295356334, −6.28754890482795448120136035087, −5.81044948300297925396078351674, −4.82357455584747164293607127474, −3.58349333689139545343733565969, −3.07194884535767220607573354520, −1.95551011048097357950164801731, −0.76050002326956211577135189994,
0.78168257758641667709171205390, 2.21805468943405894092672548246, 3.12116314339421043961747882876, 4.16390561758571625929427497781, 4.73464997332110476095770279735, 5.53499577113578137528663746053, 6.37812255881810030551506205281, 7.23298008421754636130585972797, 7.913512423333454959833899112727, 8.964396524402638073724283146640
