L(s) = 1 | + (−3.63 + 4.33i)2-s + (−5.51 − 31.5i)4-s − 80.5i·5-s − 216. i·7-s + (156. + 90.8i)8-s + (348. + 293. i)10-s − 153.·11-s − 945.·13-s + (938. + 789. i)14-s + (−963. + 347. i)16-s + 2.18e3i·17-s − 719. i·19-s + (−2.53e3 + 443. i)20-s + (558. − 664. i)22-s + 2.62e3·23-s + ⋯ |
L(s) = 1 | + (−0.643 + 0.765i)2-s + (−0.172 − 0.985i)4-s − 1.44i·5-s − 1.67i·7-s + (0.864 + 0.501i)8-s + (1.10 + 0.926i)10-s − 0.382·11-s − 1.55·13-s + (1.28 + 1.07i)14-s + (−0.940 + 0.339i)16-s + 1.83i·17-s − 0.457i·19-s + (−1.41 + 0.248i)20-s + (0.245 − 0.292i)22-s + 1.03·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0356777 - 0.411282i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0356777 - 0.411282i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.63 - 4.33i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 80.5iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 216. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 153.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 945.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.18e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 719. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 155. iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.90e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 114.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.35e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.31e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 9.40e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.96e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.32e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.34e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.40e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.86e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.19e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 8.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.68e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 2.82e4T + 8.58e9T^{2} \) |
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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
−12.54368284611497369389364180495, −10.78171384759465788259106542295, −9.997372285650643615629759062509, −8.847997766032338867551305697459, −7.84045207372581138650463266953, −6.90665851658602142352984054944, −5.22616822611872068806761953269, −4.34016243668438449354999304534, −1.37995032228404282001179454752, −0.19898474730670339305442047665,
2.46088327117755660438746606687, 2.88028317147964214733992774591, 5.10993432733991362851963680515, 6.83863312342791113844352325068, 7.84015377522568902708708093441, 9.299765642069981197726634241412, 9.955827936579738494919302649448, 11.29958903036581308005134901463, 11.84077654990929566247569398401, 12.92608003650598158639953016540