| L(s) = 1 | − 4·2-s + 16·4-s + 14·5-s − 64·8-s − 81·9-s − 56·10-s − 240i·13-s + 256·16-s + 480i·17-s + 324·18-s + 224·20-s − 429·25-s + 960i·26-s − 1.68e3i·29-s − 1.02e3·32-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 0.560·5-s − 8-s − 9-s − 0.560·10-s − 1.42i·13-s + 16-s + 1.66i·17-s + 18-s + 0.560·20-s − 0.686·25-s + 1.42i·26-s − 1.99i·29-s − 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.903 + 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2962013305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2962013305\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 41 | \( 1 + (1.51e3 - 720i)T \) |
| good | 3 | \( 1 + 81T^{2} \) |
| 5 | \( 1 - 14T + 625T^{2} \) |
| 7 | \( 1 + 2.40e3T^{2} \) |
| 11 | \( 1 + 1.46e4T^{2} \) |
| 13 | \( 1 + 240iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 480iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 1.30e5T^{2} \) |
| 23 | \( 1 - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.68e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 9.23e5T^{2} \) |
| 37 | \( 1 + 2.16e3T + 1.87e6T^{2} \) |
| 43 | \( 1 - 3.41e6T^{2} \) |
| 47 | \( 1 + 4.87e6T^{2} \) |
| 53 | \( 1 + 5.04e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.21e7T^{2} \) |
| 61 | \( 1 + 6.95e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.54e7T^{2} \) |
| 73 | \( 1 + 1.44e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.24e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.87e4iT - 8.85e7T^{2} \) |
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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
−11.49051169917892777840272912019, −10.48889056071236418047090003162, −9.800954971588268725163091697289, −8.490318423099431330109187619813, −7.939425638701714590311501716604, −6.31812317164349689255038562760, −5.59951820810910098489085300036, −3.30137606874165206178683922708, −1.91142141695570175259475171900, −0.14448920915642909908543719449,
1.71563756104478521507851507905, 3.05725846453761676013542335522, 5.20217020016027327227471323874, 6.45938034824675917137142452350, 7.39180717915875479502852447171, 8.862619056673247085625394703473, 9.267339514387753967441047743248, 10.48655479206905890769496439481, 11.49245555791443824268279490407, 12.15000576741899867884905807256
