| L(s) = 1 | + (0.882 + 2.71i)2-s + (3.14 + 4.13i)3-s + (−0.124 + 0.0905i)4-s + (−4.91 − 1.59i)5-s + (−8.45 + 12.1i)6-s + (−9.45 − 13.0i)7-s + (18.1 + 13.1i)8-s + (−7.20 + 26.0i)9-s − 14.7i·10-s + (5.09 − 36.1i)11-s + (−0.766 − 0.230i)12-s + (16.5 − 5.38i)13-s + (26.9 − 37.1i)14-s + (−8.86 − 25.3i)15-s + (−20.1 + 62.0i)16-s + (30.8 − 94.8i)17-s + ⋯ |
| L(s) = 1 | + (0.311 + 0.960i)2-s + (0.605 + 0.795i)3-s + (−0.0155 + 0.0113i)4-s + (−0.439 − 0.142i)5-s + (−0.575 + 0.829i)6-s + (−0.510 − 0.702i)7-s + (0.801 + 0.581i)8-s + (−0.266 + 0.963i)9-s − 0.466i·10-s + (0.139 − 0.990i)11-s + (−0.0184 − 0.00554i)12-s + (0.353 − 0.114i)13-s + (0.515 − 0.709i)14-s + (−0.152 − 0.436i)15-s + (−0.314 + 0.969i)16-s + (0.439 − 1.35i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0961 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0961 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.22408 + 1.11155i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22408 + 1.11155i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-3.14 - 4.13i)T \) |
| 11 | \( 1 + (-5.09 + 36.1i)T \) |
| good | 2 | \( 1 + (-0.882 - 2.71i)T + (-6.47 + 4.70i)T^{2} \) |
| 5 | \( 1 + (4.91 + 1.59i)T + (101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (9.45 + 13.0i)T + (-105. + 326. i)T^{2} \) |
| 13 | \( 1 + (-16.5 + 5.38i)T + (1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-30.8 + 94.8i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (27.1 - 37.3i)T + (-2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 - 113. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (110. - 80.2i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-26.9 - 82.9i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (216. - 157. i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (21.4 + 15.5i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 487. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (35.3 - 48.6i)T + (-3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (472. - 153. i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-432. - 594. i)T + (-6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-817. - 265. i)T + (1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 507.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-980. - 318. i)T + (2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (173. + 238. i)T + (-1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (576. - 187. i)T + (3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (39.9 - 122. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 811. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-69.6 - 214. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
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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
−16.14670509277157868643369354292, −15.58375392194610470824928088742, −14.19625654713594548319513105975, −13.57901333647607346895476024681, −11.37394897764179703698020526578, −10.11412675461939580212717452154, −8.486609720133468488357472475558, −7.21788849356127462130078610628, −5.42716465410906878390384593736, −3.68068229138091419773656903868,
2.11557729814520162507603254271, 3.73833991731603024143899330854, 6.58924726312105299894836999576, 8.034726957932576363152373308515, 9.669095098853435568433315989832, 11.31557703071746767838080547096, 12.50181240966538780418874911640, 12.95475401502985109727071797666, 14.60103589417527251015437639269, 15.68821193695993796530800470607
