| L(s) = 1 | + (−1.34 − 1.09i)3-s − 0.320·5-s + 1.42i·7-s + (0.616 + 2.93i)9-s − 3.91i·11-s + i·13-s + (0.430 + 0.349i)15-s + 1.59i·17-s + 6.48·19-s + (1.55 − 1.90i)21-s − 7.01·23-s − 4.89·25-s + (2.37 − 4.62i)27-s + 1.36·29-s − 0.176i·31-s + ⋯ |
| L(s) = 1 | + (−0.776 − 0.630i)3-s − 0.143·5-s + 0.536i·7-s + (0.205 + 0.978i)9-s − 1.18i·11-s + 0.277i·13-s + (0.111 + 0.0903i)15-s + 0.387i·17-s + 1.48·19-s + (0.338 − 0.416i)21-s − 1.46·23-s − 0.979·25-s + (0.457 − 0.889i)27-s + 0.253·29-s − 0.0317i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.024727089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.024727089\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.34 + 1.09i)T \) |
| 13 | \( 1 - iT \) |
| good | 5 | \( 1 + 0.320T + 5T^{2} \) |
| 7 | \( 1 - 1.42iT - 7T^{2} \) |
| 11 | \( 1 + 3.91iT - 11T^{2} \) |
| 17 | \( 1 - 1.59iT - 17T^{2} \) |
| 19 | \( 1 - 6.48T + 19T^{2} \) |
| 23 | \( 1 + 7.01T + 23T^{2} \) |
| 29 | \( 1 - 1.36T + 29T^{2} \) |
| 31 | \( 1 + 0.176iT - 31T^{2} \) |
| 37 | \( 1 - 3.85iT - 37T^{2} \) |
| 41 | \( 1 - 3.41iT - 41T^{2} \) |
| 43 | \( 1 + 0.549T + 43T^{2} \) |
| 47 | \( 1 - 3.22T + 47T^{2} \) |
| 53 | \( 1 + 9.10T + 53T^{2} \) |
| 59 | \( 1 - 4.21iT - 59T^{2} \) |
| 61 | \( 1 - 12.0iT - 61T^{2} \) |
| 67 | \( 1 - 3.04T + 67T^{2} \) |
| 71 | \( 1 - 9.27T + 71T^{2} \) |
| 73 | \( 1 + 3.86T + 73T^{2} \) |
| 79 | \( 1 - 13.4iT - 79T^{2} \) |
| 83 | \( 1 + 16.0iT - 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 14.1T + 97T^{2} \) |
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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
−8.878858566305161911248762232183, −8.054933164278498691295346810227, −7.57784944619818779329125330031, −6.50528082988117361312700424654, −5.88781234529993581789900617748, −5.38175683843343544829182408938, −4.27160938704759103713172327321, −3.21846058974707591212178425440, −2.09346349476507329230897124778, −0.948022546603675265730961599856,
0.48028124025015332154998355990, 1.90845889444715643932946238467, 3.37064968171408888512259882800, 4.11246273974006319459468722496, 4.88371625358313126213697054024, 5.62086988930349695318018874451, 6.47610780950258600119781084470, 7.37579086723663231485202682732, 7.83616451111123973215695962845, 9.118403714019976149704194678023
