| L(s) = 1 | + (−2.02 + 0.461i)2-s + (−16.6 − 3.80i)3-s + (−10.5 + 5.07i)4-s + (21.9 − 17.4i)5-s + 35.5·6-s + 46.4i·7-s + (44.9 − 35.8i)8-s + (190. + 91.7i)9-s + (−36.2 + 45.5i)10-s + (−90.6 − 43.6i)11-s + (194. − 44.4i)12-s + (12.8 + 16.1i)13-s + (−21.4 − 94.0i)14-s + (−432. + 208. i)15-s + (42.2 − 52.9i)16-s + (205. − 257. i)17-s + ⋯ |
| L(s) = 1 | + (−0.505 + 0.115i)2-s + (−1.85 − 0.422i)3-s + (−0.658 + 0.317i)4-s + (0.876 − 0.699i)5-s + 0.986·6-s + 0.948i·7-s + (0.702 − 0.559i)8-s + (2.35 + 1.13i)9-s + (−0.362 + 0.455i)10-s + (−0.749 − 0.360i)11-s + (1.35 − 0.308i)12-s + (0.0763 + 0.0956i)13-s + (−0.109 − 0.480i)14-s + (−1.92 + 0.924i)15-s + (0.164 − 0.206i)16-s + (0.709 − 0.889i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.574100 + 0.0968548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.574100 + 0.0968548i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-944. + 1.58e3i)T \) |
| good | 2 | \( 1 + (2.02 - 0.461i)T + (14.4 - 6.94i)T^{2} \) |
| 3 | \( 1 + (16.6 + 3.80i)T + (72.9 + 35.1i)T^{2} \) |
| 5 | \( 1 + (-21.9 + 17.4i)T + (139. - 609. i)T^{2} \) |
| 7 | \( 1 - 46.4iT - 2.40e3T^{2} \) |
| 11 | \( 1 + (90.6 + 43.6i)T + (9.12e3 + 1.14e4i)T^{2} \) |
| 13 | \( 1 + (-12.8 - 16.1i)T + (-6.35e3 + 2.78e4i)T^{2} \) |
| 17 | \( 1 + (-205. + 257. i)T + (-1.85e4 - 8.14e4i)T^{2} \) |
| 19 | \( 1 + (-236. - 492. i)T + (-8.12e4 + 1.01e5i)T^{2} \) |
| 23 | \( 1 + (-569. - 274. i)T + (1.74e5 + 2.18e5i)T^{2} \) |
| 29 | \( 1 + (-299. + 68.2i)T + (6.37e5 - 3.06e5i)T^{2} \) |
| 31 | \( 1 + (-62.5 - 274. i)T + (-8.32e5 + 4.00e5i)T^{2} \) |
| 37 | \( 1 + 845. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-671. - 2.94e3i)T + (-2.54e6 + 1.22e6i)T^{2} \) |
| 47 | \( 1 + (-3.10e3 + 1.49e3i)T + (3.04e6 - 3.81e6i)T^{2} \) |
| 53 | \( 1 + (59.0 - 73.9i)T + (-1.75e6 - 7.69e6i)T^{2} \) |
| 59 | \( 1 + (1.88e3 - 2.36e3i)T + (-2.69e6 - 1.18e7i)T^{2} \) |
| 61 | \( 1 + (1.54e3 + 353. i)T + (1.24e7 + 6.00e6i)T^{2} \) |
| 67 | \( 1 + (-635. + 306. i)T + (1.25e7 - 1.57e7i)T^{2} \) |
| 71 | \( 1 + (-429. - 891. i)T + (-1.58e7 + 1.98e7i)T^{2} \) |
| 73 | \( 1 + (1.67e3 - 1.33e3i)T + (6.31e6 - 2.76e7i)T^{2} \) |
| 79 | \( 1 - 2.44e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-1.71e3 + 7.51e3i)T + (-4.27e7 - 2.05e7i)T^{2} \) |
| 89 | \( 1 + (2.75e3 + 628. i)T + (5.65e7 + 2.72e7i)T^{2} \) |
| 97 | \( 1 + (-1.29e4 - 6.25e3i)T + (5.51e7 + 6.92e7i)T^{2} \) |
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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
−15.98703109648378146990188604092, −13.62888423497055681173001359118, −12.70554318842450694359699332067, −11.88841912661328652413848254674, −10.34601872830753361034668326419, −9.253886507021999560781984874541, −7.58059270388043679126003234088, −5.74429069218477264995061282467, −5.11094246735557534658018874352, −1.04487399080681663275166125325,
0.797659967873435222747976136504, 4.62112049313359947533669746376, 5.77968497661136795636165370567, 7.16550930672130974278343407535, 9.604368370722654342016672991169, 10.52742347376904226004597408536, 10.86867395922399435220526440201, 12.71340582603499184712245154443, 13.85895882503053190681226912184, 15.37612182310057684613657777596
