| L(s) = 1 | + (4.94 − 8.57i)5-s + (−14 − 24.2i)11-s − 4.24·13-s + (−24.7 − 42.8i)17-s + (−2.82 + 4.89i)19-s + (56 − 96.9i)23-s + (13.4 + 23.3i)25-s − 154·29-s + (−16.9 − 29.3i)31-s + (10 − 17.3i)37-s + 168.·41-s − 76·43-s + (−217. + 377. i)47-s + (−266 − 460. i)53-s − 277.·55-s + ⋯ |
| L(s) = 1 | + (0.442 − 0.766i)5-s + (−0.383 − 0.664i)11-s − 0.0905·13-s + (−0.353 − 0.611i)17-s + (−0.0341 + 0.0591i)19-s + (0.507 − 0.879i)23-s + (0.107 + 0.187i)25-s − 0.986·29-s + (−0.0983 − 0.170i)31-s + (0.0444 − 0.0769i)37-s + 0.641·41-s − 0.269·43-s + (−0.675 + 1.17i)47-s + (−0.689 − 1.19i)53-s − 0.679·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5403889151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5403889151\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-4.94 + 8.57i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (14 + 24.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 4.24T + 2.19e3T^{2} \) |
| 17 | \( 1 + (24.7 + 42.8i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (2.82 - 4.89i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-56 + 96.9i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 154T + 2.43e4T^{2} \) |
| 31 | \( 1 + (16.9 + 29.3i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-10 + 17.3i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 168.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 76T + 7.95e4T^{2} \) |
| 47 | \( 1 + (217. - 377. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (266 + 460. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (158. + 274. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (84.1 - 145. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-186 - 322. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 168T + 3.57e5T^{2} \) |
| 73 | \( 1 + (126. + 219. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-32 + 55.4i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 673.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (212. - 368. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−8.590889918189456131671816310343, −7.82214649735159320536283705037, −6.88476604518415435101460314067, −5.98114603984861458464708501148, −5.21097264884838508410363675482, −4.52537459475185058496732806578, −3.34062926136797981645300060115, −2.33701948239915661536898593873, −1.17735976574454756848259589093, −0.11248875241450074052259282687,
1.55128178279533138775699662572, 2.45380084984097101435090343821, 3.39616486040457414155815898072, 4.44361686699766475993194011883, 5.40109150961620805937903122313, 6.21545553390851700507348011175, 7.02643939492257390778192696508, 7.64056544791188118377171178393, 8.621458468534215343006897584534, 9.492411124196299840479529215117
