L(s) = 1 | + 3-s + 5-s + (−2.09 + 3.62i)7-s + 9-s + (−1.91 + 3.32i)11-s + (3.12 + 5.41i)13-s + 15-s + (−0.293 − 0.508i)17-s + (−1.03 − 1.78i)19-s + (−2.09 + 3.62i)21-s + (−0.208 − 0.360i)23-s + 25-s + 27-s + (−2.56 + 4.43i)29-s + (0.705 − 1.22i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + (−0.790 + 1.36i)7-s + 0.333·9-s + (−0.578 + 1.00i)11-s + (0.866 + 1.50i)13-s + 0.258·15-s + (−0.0712 − 0.123i)17-s + (−0.236 − 0.409i)19-s + (−0.456 + 0.790i)21-s + (−0.0434 − 0.0751i)23-s + 0.200·25-s + 0.192·27-s + (−0.475 + 0.823i)29-s + (0.126 − 0.219i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.552144258\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.552144258\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-8.13 - 0.936i)T \) |
good | 7 | \( 1 + (2.09 - 3.62i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.91 - 3.32i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.12 - 5.41i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.293 + 0.508i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.03 + 1.78i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.208 + 0.360i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.56 - 4.43i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.705 + 1.22i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.95 + 6.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.442 - 0.766i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 4.05T + 43T^{2} \) |
| 47 | \( 1 + (-0.533 + 0.923i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 6.55T + 53T^{2} \) |
| 59 | \( 1 - 7.65T + 59T^{2} \) |
| 61 | \( 1 + (2.87 + 4.98i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (7.60 - 13.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (5.89 + 10.2i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.23 - 3.87i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.17 + 2.03i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 + (3.07 + 5.32i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.940091794802973237414144238229, −8.228116931120811582414899178361, −7.10068788583720837145820532856, −6.65466387958527682278381926476, −5.82427534743702442292883232433, −5.05458289601253609294468371052, −4.14366899897317377078903136487, −3.18383891996836599248796884077, −2.28770106150961661033001932728, −1.76545210617723392952948960036,
0.38813601048778277540884227200, 1.40944051544625993634211205033, 2.85673982296275984742654546793, 3.38283485434430980931575664755, 4.06538103980287823632555833803, 5.23529417478803129416656367202, 6.02247254083225582252675015682, 6.62479094963331501149029906324, 7.57093061366329648733619696312, 8.137175809491886883989540229381