L(s) = 1 | + i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−0.887 + 2.05i)5-s + (−0.707 − 0.707i)6-s + (−1.84 + 1.84i)7-s − i·8-s − 1.00i·9-s + (−2.05 − 0.887i)10-s − 2.12i·11-s + (0.707 − 0.707i)12-s − 5.07i·13-s + (−1.84 − 1.84i)14-s + (−0.823 − 2.07i)15-s + 16-s − 2.93·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (−0.397 + 0.917i)5-s + (−0.288 − 0.288i)6-s + (−0.697 + 0.697i)7-s − 0.353i·8-s − 0.333i·9-s + (−0.648 − 0.280i)10-s − 0.639i·11-s + (0.204 − 0.204i)12-s − 1.40i·13-s + (−0.492 − 0.492i)14-s + (−0.212 − 0.536i)15-s + 0.250·16-s − 0.711·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 + 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3874885958\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3874885958\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (0.887 - 2.05i)T \) |
| 37 | \( 1 + (5.46 - 2.66i)T \) |
good | 7 | \( 1 + (1.84 - 1.84i)T - 7iT^{2} \) |
| 11 | \( 1 + 2.12iT - 11T^{2} \) |
| 13 | \( 1 + 5.07iT - 13T^{2} \) |
| 17 | \( 1 + 2.93T + 17T^{2} \) |
| 19 | \( 1 + (2.28 - 2.28i)T - 19iT^{2} \) |
| 23 | \( 1 + 5.76iT - 23T^{2} \) |
| 29 | \( 1 + (-5.76 - 5.76i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.38 + 1.38i)T - 31iT^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + 2.00iT - 43T^{2} \) |
| 47 | \( 1 + (-7.27 + 7.27i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.286 + 0.286i)T + 53iT^{2} \) |
| 59 | \( 1 + (-8.25 + 8.25i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.44 + 2.44i)T - 61iT^{2} \) |
| 67 | \( 1 + (9.89 + 9.89i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.69T + 71T^{2} \) |
| 73 | \( 1 + (-5.92 + 5.92i)T - 73iT^{2} \) |
| 79 | \( 1 + (5.73 - 5.73i)T - 79iT^{2} \) |
| 83 | \( 1 + (7.41 + 7.41i)T + 83iT^{2} \) |
| 89 | \( 1 + (11.9 + 11.9i)T + 89iT^{2} \) |
| 97 | \( 1 + 7.94T + 97T^{2} \) |
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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
−9.926732196945060197313801146143, −8.660904330345204075550372313053, −8.279951564727496054501917660059, −7.03495057546071594560683827780, −6.34551999483115206130858309054, −5.75438495389529667366999163988, −4.69765847874273107103577457520, −3.51365720762610695061361972443, −2.75423825017637774567884624824, −0.20069394796315801051191386275,
1.18840991045606660790920163743, 2.37128199322287951337213737594, 4.02583649760473708160298605596, 4.37839918265468501440486002187, 5.52384182124640917012379272820, 6.73896598779751588634961146537, 7.30508329265178798978896156958, 8.481701864040425423590117691942, 9.205354692927187737789572523050, 9.906918570971096193234159354125