L(s) = 1 | + (−0.453 + 0.891i)2-s + (−0.587 − 0.809i)4-s + (0.891 + 0.453i)5-s + (1.39 − 1.39i)7-s + (0.987 − 0.156i)8-s + (−0.809 + 0.587i)10-s + (−0.863 + 0.280i)11-s + (0.610 + 1.87i)14-s + (−0.309 + 0.951i)16-s + (−0.156 − 0.987i)20-s + (0.142 − 0.896i)22-s + (0.587 + 0.809i)25-s + (−1.95 − 0.309i)28-s + (1.14 − 0.831i)29-s + (−0.951 − 0.690i)31-s + (−0.707 − 0.707i)32-s + ⋯ |
L(s) = 1 | + (−0.453 + 0.891i)2-s + (−0.587 − 0.809i)4-s + (0.891 + 0.453i)5-s + (1.39 − 1.39i)7-s + (0.987 − 0.156i)8-s + (−0.809 + 0.587i)10-s + (−0.863 + 0.280i)11-s + (0.610 + 1.87i)14-s + (−0.309 + 0.951i)16-s + (−0.156 − 0.987i)20-s + (0.142 − 0.896i)22-s + (0.587 + 0.809i)25-s + (−1.95 − 0.309i)28-s + (1.14 − 0.831i)29-s + (−0.951 − 0.690i)31-s + (−0.707 − 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.128739174\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.128739174\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.453 - 0.891i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.891 - 0.453i)T \) |
good | 7 | \( 1 + (-1.39 + 1.39i)T - iT^{2} \) |
| 11 | \( 1 + (0.863 - 0.280i)T + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (-1.14 + 0.831i)T + (0.309 - 0.951i)T^{2} \) |
| 31 | \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (0.297 - 1.87i)T + (-0.951 - 0.309i)T^{2} \) |
| 59 | \( 1 + (-0.550 + 1.69i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (1.59 - 0.253i)T + (0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.0489 - 0.309i)T + (-0.951 - 0.309i)T^{2} \) |
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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
−9.633829988602541969685357977746, −8.520269969780796695104616670317, −7.81965166358317350108064205777, −7.29438020073792802797802480208, −6.51409662070287077383088486130, −5.51642415111702814042072818980, −4.85406099015248081162855923209, −4.01624611759756764051330815572, −2.29933680339848666106637461299, −1.19697753960302510219422727801,
1.46989387030480166228674942212, 2.22683579911069419632177290579, 3.06994509981760436281165850598, 4.67519957074371617925271394331, 5.14189762986454172655481889055, 5.89894711437990744011302639449, 7.30700927282015821633845003879, 8.376907009634480292237938633674, 8.581344404508217454273154859475, 9.307019706175796816464736363374