| L(s) = 1 | + (−8.89 − 1.37i)3-s + (−1.59 − 4.38i)5-s + (12.9 + 73.5i)7-s + (77.2 + 24.3i)9-s + (54.6 − 150. i)11-s + (−161. − 135. i)13-s + (8.18 + 41.2i)15-s + (−295. − 170. i)17-s + (−169. − 293. i)19-s + (−14.5 − 672. i)21-s + (946. + 166. i)23-s + (462. − 387. i)25-s + (−653. − 322. i)27-s + (−502. − 599. i)29-s + (226. − 1.28e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.988 − 0.152i)3-s + (−0.0638 − 0.175i)5-s + (0.264 + 1.50i)7-s + (0.953 + 0.301i)9-s + (0.451 − 1.24i)11-s + (−0.958 − 0.804i)13-s + (0.0363 + 0.183i)15-s + (−1.02 − 0.589i)17-s + (−0.469 − 0.813i)19-s + (−0.0329 − 1.52i)21-s + (1.78 + 0.315i)23-s + (0.739 − 0.620i)25-s + (−0.896 − 0.442i)27-s + (−0.597 − 0.712i)29-s + (0.235 − 1.33i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0515 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.595297 - 0.626790i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.595297 - 0.626790i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (8.89 + 1.37i)T \) |
| good | 5 | \( 1 + (1.59 + 4.38i)T + (-478. + 401. i)T^{2} \) |
| 7 | \( 1 + (-12.9 - 73.5i)T + (-2.25e3 + 821. i)T^{2} \) |
| 11 | \( 1 + (-54.6 + 150. i)T + (-1.12e4 - 9.41e3i)T^{2} \) |
| 13 | \( 1 + (161. + 135. i)T + (4.95e3 + 2.81e4i)T^{2} \) |
| 17 | \( 1 + (295. + 170. i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (169. + 293. i)T + (-6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-946. - 166. i)T + (2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (502. + 599. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-226. + 1.28e3i)T + (-8.67e5 - 3.15e5i)T^{2} \) |
| 37 | \( 1 + (611. - 1.05e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (11.6 - 13.8i)T + (-4.90e5 - 2.78e6i)T^{2} \) |
| 43 | \( 1 + (1.61e3 + 588. i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-3.75e3 + 661. i)T + (4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 - 850. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (1.07e3 + 2.95e3i)T + (-9.28e6 + 7.78e6i)T^{2} \) |
| 61 | \( 1 + (630. + 3.57e3i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-3.08e3 - 2.59e3i)T + (3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (-3.90e3 - 2.25e3i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (2.20e3 + 3.81e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (6.84e3 - 5.74e3i)T + (6.76e6 - 3.83e7i)T^{2} \) |
| 83 | \( 1 + (715. + 852. i)T + (-8.24e6 + 4.67e7i)T^{2} \) |
| 89 | \( 1 + (-4.21e3 + 2.43e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (1.29e4 + 4.70e3i)T + (6.78e7 + 5.69e7i)T^{2} \) |
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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
−12.56684806820280092626098463152, −11.59466450518876336864653213371, −10.99140009204509956429869936887, −9.409913040117524408746984422348, −8.441794911971420261601226530914, −6.87507263696852660192543471058, −5.70169187198458885988075499979, −4.82591431691060632339552442191, −2.57664097029028913549858944824, −0.46288432936274672232661020199,
1.45091755599166005876393808667, 4.09363750430182478059731503004, 4.89734487593129717071217341566, 6.94199746356394310911142847826, 7.08191383321679962960338589498, 9.175015216921376309055879889777, 10.42309015991942333532977922837, 10.92904164681206076712047773079, 12.21602049665371877558797410317, 13.02436898416146522402499186458
