L(s) = 1 | + (−0.642 + 0.766i)2-s + (1.45 − 0.943i)3-s + (−0.173 − 0.984i)4-s + (0.998 − 0.837i)5-s + (−0.211 + 1.71i)6-s + (1.20 + 2.35i)7-s + (0.866 + 0.500i)8-s + (1.22 − 2.74i)9-s + 1.30i·10-s + (0.519 − 0.618i)11-s + (−1.18 − 1.26i)12-s + (−0.540 + 1.48i)13-s + (−2.57 − 0.589i)14-s + (0.660 − 2.15i)15-s + (−0.939 + 0.342i)16-s + 1.47·17-s + ⋯ |
L(s) = 1 | + (−0.454 + 0.541i)2-s + (0.838 − 0.544i)3-s + (−0.0868 − 0.492i)4-s + (0.446 − 0.374i)5-s + (−0.0862 + 0.701i)6-s + (0.455 + 0.890i)7-s + (0.306 + 0.176i)8-s + (0.406 − 0.913i)9-s + 0.412i·10-s + (0.156 − 0.186i)11-s + (−0.340 − 0.365i)12-s + (−0.149 + 0.411i)13-s + (−0.689 − 0.157i)14-s + (0.170 − 0.557i)15-s + (−0.234 + 0.0855i)16-s + 0.358·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.000404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.000404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58562 + 0.000321009i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58562 + 0.000321009i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 3 | \( 1 + (-1.45 + 0.943i)T \) |
| 7 | \( 1 + (-1.20 - 2.35i)T \) |
good | 5 | \( 1 + (-0.998 + 0.837i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (-0.519 + 0.618i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.540 - 1.48i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 + 0.568iT - 19T^{2} \) |
| 23 | \( 1 + (-2.42 + 6.66i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.907 + 2.49i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.88 + 0.509i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.27 - 2.21i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.20 + 1.53i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.31 - 7.47i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (0.424 - 2.40i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-7.73 - 4.46i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (13.2 + 4.81i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.78 - 1.19i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (8.72 - 7.31i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (11.3 - 6.56i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.3 + 5.97i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.98 + 7.53i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.99 - 2.54i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 - 1.92T + 89T^{2} \) |
| 97 | \( 1 + (15.6 + 2.76i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.44909784763234350643776690717, −10.08364778276924464501805209599, −9.159893258992558649523401236077, −8.653137412745922804764722770682, −7.80491621613632674532400035638, −6.71771850994871969237683793674, −5.79137936558692867299781706394, −4.56422436641373923184831109272, −2.75339873221592971203973740058, −1.48158325888804130174091573297,
1.67245625768392003038949220453, 3.05659784748490184428995567556, 4.04651523669016732843932257870, 5.26770263895909019208334733802, 7.01713523045912014706289691511, 7.79369647452945504496114686388, 8.724929498009798534165876583753, 9.757098461749362719104833929218, 10.29234671750370650861592019834, 11.01183214655090338877745840453