| L(s) = 1 | + (9.74 + 16.8i)5-s + (−18.4 + 1.67i)7-s + (15.1 + 8.72i)11-s − 16.5i·13-s + (−68.6 + 118. i)17-s + (−20.6 + 11.9i)19-s + (108. − 62.8i)23-s + (−127. + 220. i)25-s − 105. i·29-s + (−95.0 − 54.8i)31-s + (−207. − 294. i)35-s + (58.5 + 101. i)37-s − 348.·41-s − 141.·43-s + (−172. − 299. i)47-s + ⋯ |
| L(s) = 1 | + (0.871 + 1.50i)5-s + (−0.995 + 0.0902i)7-s + (0.414 + 0.239i)11-s − 0.352i·13-s + (−0.979 + 1.69i)17-s + (−0.249 + 0.144i)19-s + (0.987 − 0.570i)23-s + (−1.01 + 1.76i)25-s − 0.673i·29-s + (−0.550 − 0.318i)31-s + (−1.00 − 1.42i)35-s + (0.260 + 0.450i)37-s − 1.32·41-s − 0.503·43-s + (−0.536 − 0.929i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.095010258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.095010258\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (18.4 - 1.67i)T \) |
| good | 5 | \( 1 + (-9.74 - 16.8i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-15.1 - 8.72i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 16.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (68.6 - 118. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (20.6 - 11.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-108. + 62.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 105. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (95.0 + 54.8i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-58.5 - 101. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (172. + 299. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (149. + 86.1i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-297. + 515. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (561. - 324. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (64.9 - 112. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 908. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (44.8 + 25.9i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-474. - 822. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-210. - 364. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 553. iT - 9.12e5T^{2} \) |
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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
−10.64884638349493946121840831017, −10.19292477728473534503187119930, −9.357418813855508003544264389689, −8.281986471179222164513213440134, −6.74591848755115654488550362482, −6.62537057371350629957213953294, −5.62550961328083780157083922908, −3.93886503558520955044972816781, −2.94889194237432404036394349866, −1.92011127612185081767117765435,
0.31947292206998930613121013872, 1.58708446004495217342751418697, 3.04612509433918388069908818594, 4.52246804394514529601072498454, 5.27848701456186269459807469610, 6.34780464382242724144713944732, 7.20787652251374410312536406981, 8.749457750140090244266054052180, 9.188195077067553053554930215412, 9.730797347768373263703464530944
