L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.648 − 2.13i)5-s + 0.999i·6-s + (1.49 + 0.865i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (−1.52 + 1.63i)10-s − 1.54·11-s + (0.866 − 0.499i)12-s + (−1.64 + 2.84i)13-s − 1.73i·14-s + (−0.508 + 2.17i)15-s + (−0.5 − 0.866i)16-s + (3.18 + 5.52i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.290 − 0.956i)5-s + 0.408i·6-s + (0.566 + 0.327i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (−0.483 + 0.516i)10-s − 0.466·11-s + (0.249 − 0.144i)12-s + (−0.456 + 0.790i)13-s − 0.462i·14-s + (−0.131 + 0.562i)15-s + (−0.125 − 0.216i)16-s + (0.773 + 1.33i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0845i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0845i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8511520967\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8511520967\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.648 + 2.13i)T \) |
| 37 | \( 1 + (5.20 - 3.14i)T \) |
good | 7 | \( 1 + (-1.49 - 0.865i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 + (1.64 - 2.84i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.18 - 5.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.40 + 1.96i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.76T + 23T^{2} \) |
| 29 | \( 1 - 0.719iT - 29T^{2} \) |
| 31 | \( 1 - 9.04iT - 31T^{2} \) |
| 41 | \( 1 + (-0.726 + 1.25i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 0.853T + 43T^{2} \) |
| 47 | \( 1 - 8.07iT - 47T^{2} \) |
| 53 | \( 1 + (-9.03 + 5.21i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.25 + 1.87i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.355 + 0.205i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.11 - 4.10i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.29 + 3.97i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 5.50iT - 73T^{2} \) |
| 79 | \( 1 + (-10.5 - 6.08i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-10.2 + 5.91i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.13 - 4.69i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−9.953742001701962590845756326803, −8.838545374713067351856955608470, −8.474578052691476519528980646155, −7.55341428886225146098581501186, −6.57833993347754957752941475066, −5.26587976536761096953061088790, −4.79487172620896486098161649403, −3.64813896980119168926499695990, −2.12884959829964056425265236370, −1.15086884225624524613838084382,
0.52869988211059583801492435289, 2.48821281711017884475246860269, 3.72718649730166377579094946779, 4.89157710115616851720197677493, 5.57833563580964428505922231597, 6.58673050980314949873981872432, 7.50506538881845982918839565812, 7.80153648355888023380102621447, 9.027523690325737367006551879974, 10.01920905875390096615398140092