| L(s) = 1 | + (0.131 − 1.40i)2-s + (−0.751 + 0.978i)3-s + (−1.96 − 0.371i)4-s + (−1.09 + 0.836i)5-s + (1.27 + 1.18i)6-s + (0.448 + 2.60i)7-s + (−0.781 + 2.71i)8-s + (0.382 + 1.42i)9-s + (1.03 + 1.64i)10-s + (5.29 − 0.696i)11-s + (1.83 − 1.64i)12-s + (−3.19 + 1.32i)13-s + (3.73 − 0.287i)14-s − 1.69i·15-s + (3.72 + 1.45i)16-s + (−3.67 + 2.12i)17-s + ⋯ |
| L(s) = 1 | + (0.0932 − 0.995i)2-s + (−0.433 + 0.565i)3-s + (−0.982 − 0.185i)4-s + (−0.487 + 0.374i)5-s + (0.522 + 0.484i)6-s + (0.169 + 0.985i)7-s + (−0.276 + 0.961i)8-s + (0.127 + 0.475i)9-s + (0.327 + 0.520i)10-s + (1.59 − 0.210i)11-s + (0.530 − 0.474i)12-s + (−0.887 + 0.367i)13-s + (0.997 − 0.0767i)14-s − 0.437i·15-s + (0.931 + 0.364i)16-s + (−0.891 + 0.514i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.797489 + 0.313263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.797489 + 0.313263i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.131 + 1.40i)T \) |
| 7 | \( 1 + (-0.448 - 2.60i)T \) |
| good | 3 | \( 1 + (0.751 - 0.978i)T + (-0.776 - 2.89i)T^{2} \) |
| 5 | \( 1 + (1.09 - 0.836i)T + (1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (-5.29 + 0.696i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (3.19 - 1.32i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (3.67 - 2.12i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.438 + 3.33i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (-1.78 - 6.66i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.342 + 0.827i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (1.28 + 2.22i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.296 + 0.227i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (-5.92 + 5.92i)T - 41iT^{2} \) |
| 43 | \( 1 + (-2.50 + 6.04i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-5.69 - 3.28i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.50 - 0.856i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (-1.38 - 10.5i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-5.00 - 0.658i)T + (58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (-9.27 + 12.0i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-2.99 - 2.99i)T + 71iT^{2} \) |
| 73 | \( 1 + (7.58 + 2.03i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (8.00 + 4.61i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.90 + 2.85i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-17.5 + 4.70i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−11.90923995427685119822293718516, −11.47749513144586474694705688349, −10.76232333507556065334144335462, −9.405570019327009595181505906129, −9.002911172141186078663128963655, −7.43677895123484523575984621565, −5.85574744565390026580008952181, −4.72674302504164279770975855841, −3.72544000415519374986407322505, −2.10962370527508000573949291730,
0.78110426977158011628769989507, 3.91358287215696610324052213143, 4.72323104748626485418529891557, 6.35805581043552509514136808391, 6.93310331689014786697027925036, 7.86608998686640025398429968119, 9.013452787795642765198049140996, 9.966821853487374025869009084668, 11.45038428211232131257276512011, 12.38946437457935043416766619593
