| L(s) = 1 | + (3.51 + 1.03i)2-s + (−1.74 + 2.01i)3-s + (4.55 + 2.92i)4-s + (0.606 − 4.21i)5-s + (−8.22 + 5.28i)6-s + (−6.05 − 13.2i)7-s + (−6.20 − 7.16i)8-s + (2.82 + 19.6i)9-s + (6.48 − 14.1i)10-s + (13.4 − 3.96i)11-s + (−13.8 + 4.07i)12-s + (−15.8 + 34.7i)13-s + (−7.60 − 52.8i)14-s + (7.45 + 8.60i)15-s + (−32.4 − 70.9i)16-s + (−38.3 + 24.6i)17-s + ⋯ |
| L(s) = 1 | + (1.24 + 0.364i)2-s + (−0.336 + 0.388i)3-s + (0.568 + 0.365i)4-s + (0.0542 − 0.377i)5-s + (−0.559 + 0.359i)6-s + (−0.327 − 0.716i)7-s + (−0.274 − 0.316i)8-s + (0.104 + 0.728i)9-s + (0.204 − 0.448i)10-s + (0.369 − 0.108i)11-s + (−0.333 + 0.0979i)12-s + (−0.338 + 0.742i)13-s + (−0.145 − 1.00i)14-s + (0.128 + 0.148i)15-s + (−0.506 − 1.10i)16-s + (−0.547 + 0.351i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 - 0.459i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.887 - 0.459i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.63786 + 0.398993i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63786 + 0.398993i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + (-104. + 33.8i)T \) |
| good | 2 | \( 1 + (-3.51 - 1.03i)T + (6.73 + 4.32i)T^{2} \) |
| 3 | \( 1 + (1.74 - 2.01i)T + (-3.84 - 26.7i)T^{2} \) |
| 5 | \( 1 + (-0.606 + 4.21i)T + (-119. - 35.2i)T^{2} \) |
| 7 | \( 1 + (6.05 + 13.2i)T + (-224. + 259. i)T^{2} \) |
| 11 | \( 1 + (-13.4 + 3.96i)T + (1.11e3 - 719. i)T^{2} \) |
| 13 | \( 1 + (15.8 - 34.7i)T + (-1.43e3 - 1.66e3i)T^{2} \) |
| 17 | \( 1 + (38.3 - 24.6i)T + (2.04e3 - 4.46e3i)T^{2} \) |
| 19 | \( 1 + (-87.3 - 56.1i)T + (2.84e3 + 6.23e3i)T^{2} \) |
| 29 | \( 1 + (125. - 80.6i)T + (1.01e4 - 2.21e4i)T^{2} \) |
| 31 | \( 1 + (155. + 179. i)T + (-4.23e3 + 2.94e4i)T^{2} \) |
| 37 | \( 1 + (42.7 + 297. i)T + (-4.86e4 + 1.42e4i)T^{2} \) |
| 41 | \( 1 + (47.7 - 332. i)T + (-6.61e4 - 1.94e4i)T^{2} \) |
| 43 | \( 1 + (-253. + 292. i)T + (-1.13e4 - 7.86e4i)T^{2} \) |
| 47 | \( 1 + 417.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-168. - 369. i)T + (-9.74e4 + 1.12e5i)T^{2} \) |
| 59 | \( 1 + (-15.6 + 34.3i)T + (-1.34e5 - 1.55e5i)T^{2} \) |
| 61 | \( 1 + (26.3 + 30.3i)T + (-3.23e4 + 2.24e5i)T^{2} \) |
| 67 | \( 1 + (-379. - 111. i)T + (2.53e5 + 1.62e5i)T^{2} \) |
| 71 | \( 1 + (-253. - 74.4i)T + (3.01e5 + 1.93e5i)T^{2} \) |
| 73 | \( 1 + (-389. - 250. i)T + (1.61e5 + 3.53e5i)T^{2} \) |
| 79 | \( 1 + (194. - 425. i)T + (-3.22e5 - 3.72e5i)T^{2} \) |
| 83 | \( 1 + (51.1 + 355. i)T + (-5.48e5 + 1.61e5i)T^{2} \) |
| 89 | \( 1 + (594. - 685. i)T + (-1.00e5 - 6.97e5i)T^{2} \) |
| 97 | \( 1 + (9.03 - 62.8i)T + (-8.75e5 - 2.57e5i)T^{2} \) |
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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
−16.82006840051335464844339415238, −16.24903231277468944591569166869, −14.77665512290950954889091352404, −13.67825410429467516443335767797, −12.68687543065049163931706009752, −11.11386617816156807004636425729, −9.449833908960835475636426798761, −7.08622248461371216291088651544, −5.37446853313398788670225333069, −4.04496963151100023536210563583,
3.11151658332859893360186192902, 5.29786422822890591809342705028, 6.77234214331223092591454824675, 9.203408644904126920655311828444, 11.27615086541253552023562458151, 12.29763205004894082157120922491, 13.20121678527447168076209481221, 14.63318664448349157696698869177, 15.53462324498275171226686141809, 17.51247625599237879668655934531
