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
−17.51247625599237879668655934531, −15.53462324498275171226686141809, −14.63318664448349157696698869177, −13.20121678527447168076209481221, −12.29763205004894082157120922491, −11.27615086541253552023562458151, −9.203408644904126920655311828444, −6.77234214331223092591454824675, −5.29786422822890591809342705028, −3.11151658332859893360186192902,
4.04496963151100023536210563583, 5.37446853313398788670225333069, 7.08622248461371216291088651544, 9.449833908960835475636426798761, 11.11386617816156807004636425729, 12.68687543065049163931706009752, 13.67825410429467516443335767797, 14.77665512290950954889091352404, 16.24903231277468944591569166869, 16.82006840051335464844339415238