L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + 3.84·7-s + (−0.499 − 0.866i)9-s + 3.95·11-s + (1.74 + 3.02i)13-s + (0.499 + 0.866i)15-s + (−0.871 + 1.51i)17-s + (2.95 − 3.20i)19-s + (1.92 − 3.32i)21-s + (−0.570 − 0.988i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + (2.15 + 3.73i)29-s + 0.0599·31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s + 1.45·7-s + (−0.166 − 0.288i)9-s + 1.19·11-s + (0.483 + 0.837i)13-s + (0.129 + 0.223i)15-s + (−0.211 + 0.366i)17-s + (0.677 − 0.735i)19-s + (0.419 − 0.726i)21-s + (−0.118 − 0.206i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s + (0.400 + 0.694i)29-s + 0.0107·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.526820522\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.526820522\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-2.95 + 3.20i)T \) |
good | 7 | \( 1 - 3.84T + 7T^{2} \) |
| 11 | \( 1 - 3.95T + 11T^{2} \) |
| 13 | \( 1 + (-1.74 - 3.02i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.871 - 1.51i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (0.570 + 0.988i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.15 - 3.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 0.0599T + 31T^{2} \) |
| 37 | \( 1 + 9.05T + 37T^{2} \) |
| 41 | \( 1 + (4.88 - 8.45i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.10 + 3.63i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.32 - 5.75i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.557 - 0.965i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.29 + 2.24i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.80 - 4.85i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.09 - 3.62i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.54 - 6.14i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.15 + 10.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.65 + 8.05i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 0.959T + 83T^{2} \) |
| 89 | \( 1 + (3.68 + 6.38i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.35 + 7.53i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.707568637390284260989357754940, −8.437514453842690252285964027032, −7.35159669490932829083737304457, −6.88137189954624145163899793601, −6.04421622654305795606548434383, −4.91022254869662169762550463707, −4.17652857362498982992814563027, −3.21845559551380485699432524506, −1.93878622625565916673968864529, −1.23507106240666238478772012524,
1.04646982657748523476922737302, 2.05221532524949253626131727493, 3.49576889904763115460974590446, 4.09022723978563222425364167685, 5.06642113101417304062870582080, 5.57695422440780381232926135424, 6.77519339705915149761816492014, 7.73446903777181140326268397796, 8.321133088263074344577200150555, 8.869790236405460170745616804386