L(s) = 1 | + (−1.23 + 1.23i)3-s + (−0.571 − 0.571i)5-s − i·7-s − 0.0594i·9-s + (4.55 + 4.55i)11-s + (−3.65 + 3.65i)13-s + 1.41·15-s + 2.47·17-s + (5.54 − 5.54i)19-s + (1.23 + 1.23i)21-s − 4.08i·23-s − 4.34i·25-s + (−3.63 − 3.63i)27-s + (−4.30 + 4.30i)29-s + 1.02·31-s + ⋯ |
L(s) = 1 | + (−0.714 + 0.714i)3-s + (−0.255 − 0.255i)5-s − 0.377i·7-s − 0.0198i·9-s + (1.37 + 1.37i)11-s + (−1.01 + 1.01i)13-s + 0.365·15-s + 0.599·17-s + (1.27 − 1.27i)19-s + (0.269 + 0.269i)21-s − 0.851i·23-s − 0.869i·25-s + (−0.699 − 0.699i)27-s + (−0.798 + 0.798i)29-s + 0.184·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.111348017\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.111348017\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 3 | \( 1 + (1.23 - 1.23i)T - 3iT^{2} \) |
| 5 | \( 1 + (0.571 + 0.571i)T + 5iT^{2} \) |
| 11 | \( 1 + (-4.55 - 4.55i)T + 11iT^{2} \) |
| 13 | \( 1 + (3.65 - 3.65i)T - 13iT^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + (-5.54 + 5.54i)T - 19iT^{2} \) |
| 23 | \( 1 + 4.08iT - 23T^{2} \) |
| 29 | \( 1 + (4.30 - 4.30i)T - 29iT^{2} \) |
| 31 | \( 1 - 1.02T + 31T^{2} \) |
| 37 | \( 1 + (-5.91 - 5.91i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.47iT - 41T^{2} \) |
| 43 | \( 1 + (-4.35 - 4.35i)T + 43iT^{2} \) |
| 47 | \( 1 + 6.47T + 47T^{2} \) |
| 53 | \( 1 + (1.94 + 1.94i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.90 - 6.90i)T + 59iT^{2} \) |
| 61 | \( 1 + (7.84 - 7.84i)T - 61iT^{2} \) |
| 67 | \( 1 + (3.88 - 3.88i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.44iT - 71T^{2} \) |
| 73 | \( 1 - 1.06iT - 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + (5.68 - 5.68i)T - 83iT^{2} \) |
| 89 | \( 1 - 2.81iT - 89T^{2} \) |
| 97 | \( 1 + 1.74T + 97T^{2} \) |
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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
−9.689541374009658068524794087289, −9.047727279561833499081335565898, −7.80195658316044412142630348374, −7.06182966883815014294457717168, −6.44862931495911437219636057129, −5.16300341886059473817560300983, −4.54309900059689317508261996889, −4.14433819907030089877993160745, −2.63159619314241931827351428860, −1.24064766997111045099347056619,
0.52577393199466286498128115805, 1.59571953658291757385419142761, 3.22878417130067718862880823837, 3.72775755076277089654546466737, 5.48113096247866188226557305426, 5.69173234305725297757492409969, 6.54625867676978609601267195698, 7.59630431474257654354097740842, 7.86065085766078873726702777940, 9.245895466934743848893103783470