| L(s) = 1 | + (−0.868 + 2.67i)2-s + (2.42 − 1.76i)3-s + (0.0853 + 0.0619i)4-s + (3.76 + 11.6i)5-s + (2.60 + 8.01i)6-s + (−5.42 − 3.94i)7-s + (−18.4 + 13.3i)8-s + (2.78 − 8.55i)9-s − 34.2·10-s + (35.7 + 7.25i)11-s + 0.316·12-s + (24.3 − 74.9i)13-s + (15.2 − 11.0i)14-s + (29.6 + 21.5i)15-s + (−19.5 − 60.0i)16-s + (−20.8 − 64.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.306 + 0.944i)2-s + (0.467 − 0.339i)3-s + (0.0106 + 0.00774i)4-s + (0.337 + 1.03i)5-s + (0.177 + 0.545i)6-s + (−0.292 − 0.212i)7-s + (−0.814 + 0.591i)8-s + (0.103 − 0.317i)9-s − 1.08·10-s + (0.980 + 0.198i)11-s + 0.00761·12-s + (0.519 − 1.59i)13-s + (0.291 − 0.211i)14-s + (0.509 + 0.370i)15-s + (−0.304 − 0.938i)16-s + (−0.297 − 0.914i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.153 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.994572 + 0.851813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.994572 + 0.851813i\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-35.7 - 7.25i)T \) |
| good | 2 | \( 1 + (0.868 - 2.67i)T + (-6.47 - 4.70i)T^{2} \) |
| 5 | \( 1 + (-3.76 - 11.6i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (5.42 + 3.94i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (-24.3 + 74.9i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (20.8 + 64.0i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (38.1 - 27.7i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 21.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + (66.1 + 48.0i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (20.7 - 63.8i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-309. - 225. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (298. - 216. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 153.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (309. - 224. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-181. + 557. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (96.6 + 70.2i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-267. - 823. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 + 884.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-208. - 640. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (234. + 170. i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (228. - 702. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (137. + 423. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 - 304.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-519. + 1.59e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−16.46762627589650070479187481369, −15.17122295322987553221975549191, −14.50744517054751310849521979924, −13.15889098985021622763892227063, −11.51994249748191779979029325179, −9.939355239608881881348946782716, −8.371176530443411161663672124821, −7.10478090520649188053477677227, −6.15332662248539272541416886165, −3.01855952416486627062960827742,
1.74163632182615716172540996269, 4.02944129016999110590153255201, 6.31394621600635504913093272810, 8.916050610681035311074687292841, 9.338557448375659251318092659108, 10.94141270318450058520898388799, 12.13209404657324045102349842916, 13.29864299466154991887080375895, 14.73241931255049926315011538831, 16.10486123207481819590788095975
