| L(s) = 1 | + 7.07i·5-s + 10.4i·7-s + (13.0 + 34.0i)11-s + 43.7i·13-s + 115.·17-s − 89.3i·19-s − 213. i·23-s + 75·25-s + 124.·29-s + 149.·31-s − 74.0·35-s + 161.·37-s − 172.·41-s − 258. i·43-s − 240. i·47-s + ⋯ |
| L(s) = 1 | + 0.632i·5-s + 0.565i·7-s + (0.358 + 0.933i)11-s + 0.932i·13-s + 1.64·17-s − 1.07i·19-s − 1.93i·23-s + 0.599·25-s + 0.794·29-s + 0.863·31-s − 0.357·35-s + 0.718·37-s − 0.655·41-s − 0.915i·43-s − 0.746i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.546336738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.546336738\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-13.0 - 34.0i)T \) |
| good | 5 | \( 1 - 7.07iT - 125T^{2} \) |
| 7 | \( 1 - 10.4iT - 343T^{2} \) |
| 13 | \( 1 - 43.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 213. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 149.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 172.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 258. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 240. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 334. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 382. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 609. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 260T + 3.00e5T^{2} \) |
| 71 | \( 1 - 17.3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 787. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.00e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 183.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 11.1iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.79e3T + 9.12e5T^{2} \) |
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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
−9.086624767140493844719379845515, −8.545247277947802341266240103948, −7.39854162391626484666222642441, −6.80024034954095760913214434472, −6.09387102051267705251542586449, −4.93826437507328443519211176221, −4.24660107194989409281476504479, −2.95907924462993947802819148264, −2.27729605181723478609908773447, −0.918506323632207175822646229596,
0.77879254464508489980220055189, 1.35127696561955395342231340934, 3.13976128509321437329012400558, 3.64899542400803887921154792300, 4.89012979637530647090508295432, 5.63652958011587168603149718664, 6.34443684272346681802755634981, 7.68926443225916733508694377822, 7.954626759113567988768682692640, 8.874283926317798732086318410932
