L(s) = 1 | + (−0.866 + 0.5i)5-s + (−0.366 − 1.36i)7-s + (−0.5 + 0.866i)11-s + (1 − i)17-s − i·19-s + (0.366 − 1.36i)23-s + (0.499 − 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.5 − 0.866i)31-s + (1 + 0.999i)35-s + (−0.5 − 0.866i)41-s + (−0.866 + 0.5i)49-s + (−1 − i)53-s − 0.999i·55-s + (−0.866 + 0.5i)59-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)5-s + (−0.366 − 1.36i)7-s + (−0.5 + 0.866i)11-s + (1 − i)17-s − i·19-s + (0.366 − 1.36i)23-s + (0.499 − 0.866i)25-s + (0.866 + 0.5i)29-s + (−0.5 − 0.866i)31-s + (1 + 0.999i)35-s + (−0.5 − 0.866i)41-s + (−0.866 + 0.5i)49-s + (−1 − i)53-s − 0.999i·55-s + (−0.866 + 0.5i)59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7880529591\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7880529591\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
good | 7 | \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 17 | \( 1 + (-1 + i)T - iT^{2} \) |
| 19 | \( 1 + iT - T^{2} \) |
| 23 | \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + (-1 - i)T + iT^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 - iT - T^{2} \) |
| 97 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
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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
−9.625197419244186576980159665421, −8.506734651418224713247825696355, −7.58924146405739417352681915240, −7.13790900736843269939375503332, −6.56651311369008780948819692533, −5.05587261847517322266761732452, −4.40052075700658960799858091049, −3.45995092856768821334485277478, −2.57351230167932303089028828760, −0.65225055960135622747187196493,
1.50796677308599097424015984281, 3.06355604173032085187074990485, 3.60193394488654963506184967820, 4.94785486995268190644900432953, 5.66744758263255317793636553171, 6.30795519151329098363724291381, 7.69379927564925472675633588040, 8.148947551716979215243398589808, 8.843509992942847662647645541250, 9.596244041133769375249498928781