| L(s) = 1 | + (−1.45 + 4.49i)2-s + (−2.42 + 1.76i)3-s + (−11.5 − 8.41i)4-s + (1.86 + 5.72i)5-s + (−4.37 − 13.4i)6-s + (−8.05 − 5.85i)7-s + (24.1 − 17.5i)8-s + (2.78 − 8.55i)9-s − 28.4·10-s + (8.30 + 35.5i)11-s + 42.9·12-s + (−27.8 + 85.8i)13-s + (38.0 − 27.6i)14-s + (−14.6 − 10.6i)15-s + (8.12 + 24.9i)16-s + (13.8 + 42.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.516 + 1.58i)2-s + (−0.467 + 0.339i)3-s + (−1.44 − 1.05i)4-s + (0.166 + 0.512i)5-s + (−0.297 − 0.916i)6-s + (−0.435 − 0.316i)7-s + (1.06 − 0.774i)8-s + (0.103 − 0.317i)9-s − 0.899·10-s + (0.227 + 0.973i)11-s + 1.03·12-s + (−0.594 + 1.83i)13-s + (0.726 − 0.527i)14-s + (−0.251 − 0.182i)15-s + (0.126 + 0.390i)16-s + (0.197 + 0.608i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0904429 - 0.652371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0904429 - 0.652371i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.42 - 1.76i)T \) |
| 11 | \( 1 + (-8.30 - 35.5i)T \) |
| good | 2 | \( 1 + (1.45 - 4.49i)T + (-6.47 - 4.70i)T^{2} \) |
| 5 | \( 1 + (-1.86 - 5.72i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (8.05 + 5.85i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (27.8 - 85.8i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-13.8 - 42.6i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-110. + 80.5i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 71.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-119. - 86.8i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-2.38 + 7.34i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (8.18 + 5.94i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (305. - 222. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 276.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-191. + 139. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (68.6 - 211. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (136. + 99.4i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-70.1 - 215. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + 362.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (219. + 676. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-845. - 614. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-15.6 + 48.3i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (378. + 1.16e3i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 - 964.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-78.8 + 242. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−16.74476756330801286789228420870, −15.93202485395890305660599778132, −14.77405595524746239593524484949, −13.90032202130256329727634722154, −11.92298002049074187242341145946, −10.08192736320502007499520639788, −9.158887765485226408929073411390, −7.21943912358481746428080179781, −6.48940066816493698980875849404, −4.71564007572031154349394851724,
0.78068424904051998672772859366, 3.09430483491226042306453162522, 5.54716308241448878575945122750, 8.072546503255865555947427481566, 9.529313964595869533964817004762, 10.57281469246551439051869529445, 11.93186066347842704589733208440, 12.59384546976179806755359388784, 13.75205953432884224624218930194, 15.91421513362443899466360986111
