L(s) = 1 | + (−0.5 − 1.53i)2-s + (0.309 − 0.951i)3-s + (−1.30 + 0.951i)4-s + 5-s − 1.61·6-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.5 − 1.53i)10-s + (0.499 + 1.53i)12-s + (0.309 − 0.951i)15-s + (−1.61 − 1.17i)17-s + (−0.5 + 1.53i)18-s + (0.190 + 0.587i)19-s + (−1.30 + 0.951i)20-s + (1.30 + 0.951i)23-s + (0.809 − 0.587i)24-s + ⋯ |
L(s) = 1 | + (−0.5 − 1.53i)2-s + (0.309 − 0.951i)3-s + (−1.30 + 0.951i)4-s + 5-s − 1.61·6-s + (0.809 + 0.587i)8-s + (−0.809 − 0.587i)9-s + (−0.5 − 1.53i)10-s + (0.499 + 1.53i)12-s + (0.309 − 0.951i)15-s + (−1.61 − 1.17i)17-s + (−0.5 + 1.53i)18-s + (0.190 + 0.587i)19-s + (−1.30 + 0.951i)20-s + (1.30 + 0.951i)23-s + (0.809 − 0.587i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7736012683\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7736012683\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 11 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 47 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 97 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
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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
−11.02789588833819225951893202583, −9.930872353262084553879721338897, −9.157029429579873685916519069702, −8.647189479365806295540937322112, −7.26146128510121315295082439118, −6.32357887775229766665561452861, −4.93952840794237129268936575334, −3.23591597422873610430619175012, −2.38076062677276187021180587857, −1.33345990569039309214587107852,
2.54558009187604436768145381121, 4.37976540661138603920334990965, 5.23984024455314915212165581436, 6.19801910734799788340793746357, 6.93056979503681191081223231883, 8.303034948489753777165551496494, 8.871238871325229142833890603863, 9.507923177516655770900507301274, 10.43532005736914966463893420513, 11.22096004681283686428956563747