| L(s) = 1 | + (−1.70 + 1.04i)2-s + (−4.04 + 2.06i)3-s + (1.80 − 3.56i)4-s + (2.33 + 4.42i)5-s + (4.73 − 7.74i)6-s + (−3.03 + 5.94i)7-s + (0.650 + 7.97i)8-s + (6.82 − 9.40i)9-s + (−8.60 − 5.09i)10-s + (−2.74 + 10.6i)11-s + (0.0352 + 18.1i)12-s + (3.87 + 24.4i)13-s + (−1.06 − 13.3i)14-s + (−18.5 − 13.0i)15-s + (−9.45 − 12.9i)16-s + (2.85 − 18.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.852 + 0.523i)2-s + (−1.34 + 0.687i)3-s + (0.452 − 0.891i)4-s + (0.466 + 0.884i)5-s + (0.789 − 1.29i)6-s + (−0.432 + 0.849i)7-s + (0.0813 + 0.996i)8-s + (0.758 − 1.04i)9-s + (−0.860 − 0.509i)10-s + (−0.249 + 0.968i)11-s + (0.00293 + 1.51i)12-s + (0.297 + 1.88i)13-s + (−0.0757 − 0.950i)14-s + (−1.23 − 0.872i)15-s + (−0.590 − 0.806i)16-s + (0.167 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.166760 - 0.355958i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.166760 - 0.355958i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.70 - 1.04i)T \) |
| 5 | \( 1 + (-2.33 - 4.42i)T \) |
| 11 | \( 1 + (2.74 - 10.6i)T \) |
| good | 3 | \( 1 + (4.04 - 2.06i)T + (5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (3.03 - 5.94i)T + (-28.8 - 39.6i)T^{2} \) |
| 13 | \( 1 + (-3.87 - 24.4i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-2.85 + 18.0i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (-10.7 - 3.50i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (20.8 + 20.8i)T + 529iT^{2} \) |
| 29 | \( 1 + (9.88 + 30.4i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (8.80 - 12.1i)T + (-296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-6.94 + 13.6i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-9.85 - 3.20i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (-14.2 + 14.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-8.10 - 15.9i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (69.7 - 11.0i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-24.7 - 76.2i)T + (-2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (22.3 + 30.8i)T + (-1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-15.3 + 15.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + (12.2 + 16.8i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (36.2 - 71.1i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-58.2 + 80.1i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-16.4 + 103. i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + 69.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-22.7 - 143. i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−12.08039445784135493604125850376, −11.53197669522780440030921799726, −10.58762850853604738784707871547, −9.698930498353820407184929991904, −9.242210767986219869394941933469, −7.37398087357159124861838907592, −6.43150992520704465260097549756, −5.80510579548140077514606598431, −4.58767280306723909573733263494, −2.22173759358695129842965041459,
0.36742727457074843363703605235, 1.28798960589435009375413002690, 3.51352026940250447860329045820, 5.44500204694035551110074929369, 6.20497758053966517295108171732, 7.55929479949825890023331673767, 8.345818518940517373729098902419, 9.782137119155541845129702594120, 10.59561960963917839017703235135, 11.25953248948852801268713677435
