| L(s) = 1 | + (−1.46 + 1.06i)2-s + (−0.927 − 2.85i)3-s + (−1.46 + 4.49i)4-s + (17.3 + 12.6i)5-s + (4.39 + 3.18i)6-s + (−5.78 + 17.8i)7-s + (−7.11 − 21.8i)8-s + (−7.28 + 5.29i)9-s − 38.8·10-s + (30.2 − 20.4i)11-s + 14.1·12-s + (−11.2 + 8.15i)13-s + (−10.4 − 32.2i)14-s + (19.8 − 61.2i)15-s + (3.09 + 2.24i)16-s + (17.6 + 12.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.517 + 0.375i)2-s + (−0.178 − 0.549i)3-s + (−0.182 + 0.562i)4-s + (1.55 + 1.12i)5-s + (0.298 + 0.217i)6-s + (−0.312 + 0.961i)7-s + (−0.314 − 0.967i)8-s + (−0.269 + 0.195i)9-s − 1.22·10-s + (0.828 − 0.560i)11-s + 0.341·12-s + (−0.239 + 0.174i)13-s + (−0.199 − 0.614i)14-s + (0.342 − 1.05i)15-s + (0.0482 + 0.0350i)16-s + (0.251 + 0.182i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.846260 + 0.578811i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.846260 + 0.578811i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.927 + 2.85i)T \) |
| 11 | \( 1 + (-30.2 + 20.4i)T \) |
| good | 2 | \( 1 + (1.46 - 1.06i)T + (2.47 - 7.60i)T^{2} \) |
| 5 | \( 1 + (-17.3 - 12.6i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (5.78 - 17.8i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (11.2 - 8.15i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-17.6 - 12.7i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (39.0 + 120. i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 1.06T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-40.3 + 124. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-156. + 113. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-12.8 + 39.7i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (18.3 + 56.3i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 + 172.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (41.3 + 127. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-32.5 + 23.6i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-62.0 + 191. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (166. + 120. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 474.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-342. - 248. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (335. - 1.03e3i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (465. - 337. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (543. + 394. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 + 311.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (553. - 402. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−16.98404301600582845784027090002, −15.29687030632507024108492697914, −13.94925983487808401875782610512, −12.98710738668344111158162369816, −11.52890568527475522925977432862, −9.810958064954590973185456868954, −8.784805533155925274538450884031, −6.88814118782750226199381244667, −6.05259101946214510209683846554, −2.66913916292836820110596499036,
1.36877990919721438725607690042, 4.75935449459932389256205528145, 6.13262318344527032510038748038, 8.783579660980889136055034433914, 9.874275094112590821944710479441, 10.31443935282298022796067247747, 12.28415073890113665748881075686, 13.70819166805364605444221137101, 14.57468051212978799600606168022, 16.53708990260723781810816940893
