| L(s) = 1 | + (−1.91 − 0.563i)2-s + (1.96 − 2.26i)3-s + (3.36 + 2.16i)4-s + (−0.966 + 6.71i)5-s + (−5.04 + 3.24i)6-s + (5.61 + 12.3i)7-s + (−5.23 − 6.04i)8-s + (−1.28 − 8.90i)9-s + (5.64 − 12.3i)10-s + (−47.7 + 14.0i)11-s + (11.5 − 3.38i)12-s + (−22.9 + 50.2i)13-s + (−3.84 − 26.7i)14-s + (13.3 + 15.3i)15-s + (6.64 + 14.5i)16-s + (18.3 − 11.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (0.378 − 0.436i)3-s + (0.420 + 0.270i)4-s + (−0.0864 + 0.601i)5-s + (−0.343 + 0.220i)6-s + (0.303 + 0.664i)7-s + (−0.231 − 0.267i)8-s + (−0.0474 − 0.329i)9-s + (0.178 − 0.390i)10-s + (−1.30 + 0.384i)11-s + (0.276 − 0.0813i)12-s + (−0.489 + 1.07i)13-s + (−0.0734 − 0.511i)14-s + (0.229 + 0.264i)15-s + (0.103 + 0.227i)16-s + (0.261 − 0.167i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.115 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.115 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.718370 + 0.639578i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.718370 + 0.639578i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.91 + 0.563i)T \) |
| 3 | \( 1 + (-1.96 + 2.26i)T \) |
| 23 | \( 1 + (-31.4 - 105. i)T \) |
| good | 5 | \( 1 + (0.966 - 6.71i)T + (-119. - 35.2i)T^{2} \) |
| 7 | \( 1 + (-5.61 - 12.3i)T + (-224. + 259. i)T^{2} \) |
| 11 | \( 1 + (47.7 - 14.0i)T + (1.11e3 - 719. i)T^{2} \) |
| 13 | \( 1 + (22.9 - 50.2i)T + (-1.43e3 - 1.66e3i)T^{2} \) |
| 17 | \( 1 + (-18.3 + 11.7i)T + (2.04e3 - 4.46e3i)T^{2} \) |
| 19 | \( 1 + (-10.1 - 6.53i)T + (2.84e3 + 6.23e3i)T^{2} \) |
| 29 | \( 1 + (219. - 140. i)T + (1.01e4 - 2.21e4i)T^{2} \) |
| 31 | \( 1 + (-146. - 168. i)T + (-4.23e3 + 2.94e4i)T^{2} \) |
| 37 | \( 1 + (-41.3 - 287. i)T + (-4.86e4 + 1.42e4i)T^{2} \) |
| 41 | \( 1 + (-41.6 + 289. i)T + (-6.61e4 - 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-107. + 123. i)T + (-1.13e4 - 7.86e4i)T^{2} \) |
| 47 | \( 1 + 90.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + (169. + 370. i)T + (-9.74e4 + 1.12e5i)T^{2} \) |
| 59 | \( 1 + (90.3 - 197. i)T + (-1.34e5 - 1.55e5i)T^{2} \) |
| 61 | \( 1 + (477. + 551. i)T + (-3.23e4 + 2.24e5i)T^{2} \) |
| 67 | \( 1 + (88.6 + 26.0i)T + (2.53e5 + 1.62e5i)T^{2} \) |
| 71 | \( 1 + (-393. - 115. i)T + (3.01e5 + 1.93e5i)T^{2} \) |
| 73 | \( 1 + (-288. - 185. i)T + (1.61e5 + 3.53e5i)T^{2} \) |
| 79 | \( 1 + (282. - 618. i)T + (-3.22e5 - 3.72e5i)T^{2} \) |
| 83 | \( 1 + (61.0 + 424. i)T + (-5.48e5 + 1.61e5i)T^{2} \) |
| 89 | \( 1 + (-309. + 357. i)T + (-1.00e5 - 6.97e5i)T^{2} \) |
| 97 | \( 1 + (-158. + 1.10e3i)T + (-8.75e5 - 2.57e5i)T^{2} \) |
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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
−12.77265797401034017972706276604, −11.85563338183418693021152098163, −10.89127452395803484937104670943, −9.763924853968497897778555026315, −8.764741350372090285358602575436, −7.64116915263203900174583346230, −6.86360032094837596435831179504, −5.20392015751500900932649039752, −3.09787860496301676003270627465, −1.90952734807124515627243853153,
0.55586905200443578302245803334, 2.72635433293212746881209823782, 4.55314113924620130074152509989, 5.75390843851668294481495882615, 7.64791030301071581324874133796, 8.086680529270905006584336126505, 9.340580836562453796354560202133, 10.36846458305275946168093720435, 11.02837627580551664703198098097, 12.59539433681911485359891040239
