L(s) = 1 | + (3.49 + 1.95i)2-s + (8.36 + 13.6i)4-s + 47.1·5-s − 67.4i·7-s + (2.56 + 63.9i)8-s + (164. + 92.0i)10-s − 69.3i·11-s + 3.28·13-s + (131. − 235. i)14-s + (−115. + 228. i)16-s − 116.·17-s + 513. i·19-s + (394. + 642. i)20-s + (135. − 242. i)22-s − 134. i·23-s + ⋯ |
L(s) = 1 | + (0.872 + 0.488i)2-s + (0.522 + 0.852i)4-s + 1.88·5-s − 1.37i·7-s + (0.0400 + 0.999i)8-s + (1.64 + 0.920i)10-s − 0.573i·11-s + 0.0194·13-s + (0.671 − 1.20i)14-s + (−0.453 + 0.891i)16-s − 0.404·17-s + 1.42i·19-s + (0.985 + 1.60i)20-s + (0.280 − 0.500i)22-s − 0.253i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(3.59033 + 1.01364i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.59033 + 1.01364i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.49 - 1.95i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 47.1T + 625T^{2} \) |
| 7 | \( 1 + 67.4iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 69.3iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 3.28T + 2.85e4T^{2} \) |
| 17 | \( 1 + 116.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 513. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 134. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 772.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.06e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.03e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.27e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 2.52e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 694. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 1.16e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 5.53e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 3.20e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.36e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 166. iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.08e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 724. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 8.15e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 6.96e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 278.T + 8.85e7T^{2} \) |
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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
−13.51937572074873002968006734978, −12.47142603374785349020388351407, −10.84158839237271287808498787973, −10.09257273834682825407164874351, −8.621975529740843084264952636400, −7.10712179654228398201985907902, −6.16491656015233738034126502368, −5.13175384282017589337715090755, −3.54316972789741444763524938943, −1.76588040584694720021243693412,
1.84783307854502259272649352812, 2.67438788513312079015128622003, 4.94799486284375279418107348312, 5.76187305478359043610863024404, 6.72192081889682693408721095077, 9.097185279337924814806116181391, 9.668887235198407015657130114100, 10.88118780254409124229773820033, 12.04939217924168236446063422430, 13.10638725259739318338506828612