| L(s) = 1 | + (−3.21 + 8.40i)3-s + 11.1i·5-s + 46.9·7-s + (−60.3 − 54.0i)9-s − 200. i·11-s − 22.3·13-s + (−93.9 − 35.9i)15-s − 344. i·17-s + 59.9·19-s + (−150. + 394. i)21-s + 212. i·23-s − 125.·25-s + (648. − 333. i)27-s − 578. i·29-s − 490.·31-s + ⋯ |
| L(s) = 1 | + (−0.356 + 0.934i)3-s + 0.447i·5-s + 0.958·7-s + (−0.745 − 0.666i)9-s − 1.66i·11-s − 0.132·13-s + (−0.417 − 0.159i)15-s − 1.19i·17-s + 0.166·19-s + (−0.342 + 0.894i)21-s + 0.402i·23-s − 0.200·25-s + (0.888 − 0.457i)27-s − 0.687i·29-s − 0.509·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.579255870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.579255870\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (3.21 - 8.40i)T \) |
| 5 | \( 1 - 11.1iT \) |
| good | 7 | \( 1 - 46.9T + 2.40e3T^{2} \) |
| 11 | \( 1 + 200. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 22.3T + 2.85e4T^{2} \) |
| 17 | \( 1 + 344. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 59.9T + 1.30e5T^{2} \) |
| 23 | \( 1 - 212. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 578. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 490.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.93e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.63e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.16e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.28e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 2.62e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 4.10e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.79e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.56e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 5.37e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.32e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.80e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 3.38e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.44e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 5.44e3T + 8.85e7T^{2} \) |
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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
−11.29573641795615461153830046230, −10.68514220030196769247283663903, −9.517593890349793500734075908600, −8.620442789171916130239725222591, −7.51224053051446989363205033780, −6.03413209820759796429306730379, −5.22359396898258179760922990034, −3.97980079576240724553351931413, −2.77219424928228006875500486936, −0.61267174170352340039287400928,
1.26064673812697185726582087600, 2.20908018567285924435845027537, 4.37281628635080703867574937395, 5.30733816288787989409039620615, 6.56371965598470347563954902336, 7.62730457032964370031712662046, 8.287798630144461171199993613661, 9.574763911548010246800317793407, 10.78394996696276037409380621257, 11.64292530801295798265944649207
