| 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
−13.10638725259739318338506828612, −12.04939217924168236446063422430, −10.88118780254409124229773820033, −9.668887235198407015657130114100, −9.097185279337924814806116181391, −6.72192081889682693408721095077, −5.76187305478359043610863024404, −4.94799486284375279418107348312, −2.67438788513312079015128622003, −1.84783307854502259272649352812,
1.76588040584694720021243693412, 3.54316972789741444763524938943, 5.13175384282017589337715090755, 6.16491656015233738034126502368, 7.10712179654228398201985907902, 8.621975529740843084264952636400, 10.09257273834682825407164874351, 10.84158839237271287808498787973, 12.47142603374785349020388351407, 13.51937572074873002968006734978
