| L(s) = 1 | + (−4.5 − 7.79i)3-s + (39.3 − 68.1i)5-s + (−40.5 + 70.1i)9-s + (−345. − 598. i)11-s − 818.·13-s − 708.·15-s + (554. + 960. i)17-s + (−286. + 496. i)19-s + (−1.25e3 + 2.18e3i)23-s + (−1.53e3 − 2.65e3i)25-s + 729·27-s − 3.25e3·29-s + (5.05e3 + 8.76e3i)31-s + (−3.11e3 + 5.38e3i)33-s + (−2.43e3 + 4.21e3i)37-s + ⋯ |
| L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.703 − 1.21i)5-s + (−0.166 + 0.288i)9-s + (−0.861 − 1.49i)11-s − 1.34·13-s − 0.812·15-s + (0.465 + 0.806i)17-s + (−0.182 + 0.315i)19-s + (−0.496 + 0.859i)23-s + (−0.490 − 0.849i)25-s + 0.192·27-s − 0.719·29-s + (0.945 + 1.63i)31-s + (−0.497 + 0.861i)33-s + (−0.292 + 0.506i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6519688496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6519688496\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (4.5 + 7.79i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-39.3 + 68.1i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (345. + 598. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 818.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-554. - 960. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (286. - 496. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.25e3 - 2.18e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.25e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-5.05e3 - 8.76e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (2.43e3 - 4.21e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.30e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.30e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-6.45e3 + 1.11e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-9.77e3 - 1.69e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.25e4 + 2.17e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.56e4 - 2.71e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.79e4 + 4.84e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 2.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (3.38e4 + 5.85e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (7.03e3 - 1.21e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 7.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-160. + 277. i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.12e5T + 8.58e9T^{2} \) |
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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
−10.07063365993961363091190051477, −9.069166137143647470921573014543, −8.277784635628010683501585053747, −7.54642164592174511153403678563, −6.15341726245543779962908989374, −5.51434185144685156469246146685, −4.79614377797507738807053062380, −3.22921224745443967011301324273, −1.93353275365786013877437808135, −0.937635321912667336934340148633,
0.16350537271178732751894324320, 2.27216269510107732894365325685, 2.68181713973450480556465538386, 4.29459992638884569441479682175, 5.13955098848657169366369657713, 6.14733292967741216363395231288, 7.15591689020391763104503948984, 7.69390171999171770428906054037, 9.347570929574025191321589420179, 9.998486741416113647930281721139
