L(s) = 1 | + (−1.06 + 0.613i)2-s + (0.819 + 0.473i)3-s + (−0.246 + 0.426i)4-s − 1.16·6-s + (1.74 + 1.00i)7-s − 3.06i·8-s + (−1.05 − 1.82i)9-s − 0.974·11-s + (−0.403 + 0.233i)12-s + (−4.65 − 2.68i)13-s − 2.47·14-s + (1.38 + 2.40i)16-s + (6.57 − 3.79i)17-s + (2.23 + 1.29i)18-s + (0.489 − 0.847i)19-s + ⋯ |
L(s) = 1 | + (−0.751 + 0.434i)2-s + (0.473 + 0.273i)3-s + (−0.123 + 0.213i)4-s − 0.474·6-s + (0.660 + 0.381i)7-s − 1.08i·8-s + (−0.350 − 0.607i)9-s − 0.293·11-s + (−0.116 + 0.0673i)12-s + (−1.29 − 0.745i)13-s − 0.662·14-s + (0.346 + 0.600i)16-s + (1.59 − 0.920i)17-s + (0.527 + 0.304i)18-s + (0.112 − 0.194i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.858995 - 0.254228i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.858995 - 0.254228i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 + (3.81 + 4.73i)T \) |
good | 2 | \( 1 + (1.06 - 0.613i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.819 - 0.473i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (-1.74 - 1.00i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 0.974T + 11T^{2} \) |
| 13 | \( 1 + (4.65 + 2.68i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-6.57 + 3.79i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.489 + 0.847i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 7.57iT - 23T^{2} \) |
| 29 | \( 1 + 3.68T + 29T^{2} \) |
| 31 | \( 1 + 1.02T + 31T^{2} \) |
| 41 | \( 1 + (4.57 - 7.91i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 4.36iT - 43T^{2} \) |
| 47 | \( 1 - 8.32iT - 47T^{2} \) |
| 53 | \( 1 + (-10.9 + 6.31i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.00 - 8.66i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.168 - 0.291i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.817 - 0.471i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.32 + 9.21i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 10.8iT - 73T^{2} \) |
| 79 | \( 1 + (-8.73 + 15.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.992 - 0.573i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.34 + 2.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 12.0iT - 97T^{2} \) |
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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
−9.806397238623248315171958675163, −9.068449400331552523985484481285, −8.318409109062837889446659454350, −7.70664745870952886795626078629, −6.94771187437686914713037011252, −5.63340037539179551476881877205, −4.75518781888917412410452081088, −3.47488079504388763664559724437, −2.58463014660098512129109009176, −0.53381087900583211373281907407,
1.45242609268195215121990957174, 2.22838648265336845210422940359, 3.61905560404980100678269109347, 5.08733039019687671679288841759, 5.52140302315301388014129473621, 7.21109172884533635625704601934, 7.83952291313148766663619904512, 8.434134475454006932639859508836, 9.442748431934453189098464655409, 10.05876476860003656516139868571