| L(s) = 1 | + (−0.139 − 1.40i)2-s + (0.298 − 1.70i)3-s + (−1.96 + 0.392i)4-s + (−2.42 + 0.426i)5-s + (−2.44 − 0.181i)6-s + (2.49 − 2.97i)7-s + (0.826 + 2.70i)8-s + (−2.82 − 1.01i)9-s + (0.938 + 3.34i)10-s + (0.0101 − 0.0573i)11-s + (0.0852 + 3.46i)12-s + (4.33 + 1.57i)13-s + (−4.53 − 3.09i)14-s + (0.00606 + 4.25i)15-s + (3.69 − 1.54i)16-s + (1.79 − 1.03i)17-s + ⋯ |
| L(s) = 1 | + (−0.0986 − 0.995i)2-s + (0.172 − 0.985i)3-s + (−0.980 + 0.196i)4-s + (−1.08 + 0.190i)5-s + (−0.997 − 0.0741i)6-s + (0.943 − 1.12i)7-s + (0.292 + 0.956i)8-s + (−0.940 − 0.339i)9-s + (0.296 + 1.05i)10-s + (0.00305 − 0.0173i)11-s + (0.0245 + 0.999i)12-s + (1.20 + 0.437i)13-s + (−1.21 − 0.828i)14-s + (0.00156 + 1.09i)15-s + (0.922 − 0.385i)16-s + (0.434 − 0.250i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.282254 - 0.814198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.282254 - 0.814198i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.139 + 1.40i)T \) |
| 3 | \( 1 + (-0.298 + 1.70i)T \) |
| good | 5 | \( 1 + (2.42 - 0.426i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.49 + 2.97i)T + (-1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.0101 + 0.0573i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.33 - 1.57i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.79 + 1.03i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.46 + 1.42i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.37 + 3.67i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.37 - 3.76i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.90 - 5.84i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.41 - 4.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.375 - 1.03i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (9.50 + 1.67i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (4.18 + 3.51i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 + (0.277 + 1.57i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.61 + 1.35i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (2.58 - 7.10i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-4.08 - 7.07i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.26 + 5.65i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.36 + 6.51i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-10.7 + 3.89i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (1.85 + 1.07i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.895 - 5.07i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−13.25697183586852480077506374298, −12.05681361720099099281567036075, −11.30555917079927258604417744866, −10.60071763506568131995226773124, −8.679142136790966691610688218589, −8.020896821307668562313472768417, −6.86914186706052107908875266204, −4.63405219688177534148195837010, −3.33000666841464786022559680785, −1.20307971278223970494996733429,
3.71954182673120912829307462177, 4.90781449165702029954518146451, 5.98194552098979495072346614123, 8.109370797443560507756220595367, 8.302244650114572092496900436762, 9.521892887793954873915351547866, 10.96960440047651766419852090522, 11.89797241004684418425063437858, 13.35476764124661650271614420180, 14.68726191838388896292151784454
