| L(s) = 1 | + (0.922 − 2.83i)2-s + (0.661 + 0.595i)3-s + (−3.97 − 2.88i)4-s + (−2.59 + 4.49i)5-s + (2.29 − 1.32i)6-s + (−0.323 + 3.08i)7-s + (−2.20 + 1.60i)8-s + (−0.858 − 8.16i)9-s + (10.3 + 11.5i)10-s + (0.573 + 1.28i)11-s + (−0.908 − 4.27i)12-s + (−4.15 + 19.5i)13-s + (8.45 + 3.76i)14-s + (−4.38 + 1.42i)15-s + (−3.55 − 10.9i)16-s + (10.7 − 24.2i)17-s + ⋯ |
| L(s) = 1 | + (0.461 − 1.41i)2-s + (0.220 + 0.198i)3-s + (−0.993 − 0.721i)4-s + (−0.518 + 0.898i)5-s + (0.383 − 0.221i)6-s + (−0.0462 + 0.440i)7-s + (−0.275 + 0.200i)8-s + (−0.0953 − 0.907i)9-s + (1.03 + 1.15i)10-s + (0.0521 + 0.117i)11-s + (−0.0757 − 0.356i)12-s + (−0.319 + 1.50i)13-s + (0.603 + 0.268i)14-s + (−0.292 + 0.0950i)15-s + (−0.222 − 0.684i)16-s + (0.635 − 1.42i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.283 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.949321 - 0.709607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.949321 - 0.709607i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 31 | \( 1 + (-11.2 - 28.8i)T \) |
| good | 2 | \( 1 + (-0.922 + 2.83i)T + (-3.23 - 2.35i)T^{2} \) |
| 3 | \( 1 + (-0.661 - 0.595i)T + (0.940 + 8.95i)T^{2} \) |
| 5 | \( 1 + (2.59 - 4.49i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (0.323 - 3.08i)T + (-47.9 - 10.1i)T^{2} \) |
| 11 | \( 1 + (-0.573 - 1.28i)T + (-80.9 + 89.9i)T^{2} \) |
| 13 | \( 1 + (4.15 - 19.5i)T + (-154. - 68.7i)T^{2} \) |
| 17 | \( 1 + (-10.7 + 24.2i)T + (-193. - 214. i)T^{2} \) |
| 19 | \( 1 + (23.5 - 5.00i)T + (329. - 146. i)T^{2} \) |
| 23 | \( 1 + (12.9 + 17.8i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (-25.3 - 8.23i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (-23.5 + 13.5i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-25.6 - 28.4i)T + (-175. + 1.67e3i)T^{2} \) |
| 43 | \( 1 + (7.11 + 33.4i)T + (-1.68e3 + 752. i)T^{2} \) |
| 47 | \( 1 + (11.2 + 34.6i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-30.3 + 3.19i)T + (2.74e3 - 584. i)T^{2} \) |
| 59 | \( 1 + (53.7 - 59.6i)T + (-363. - 3.46e3i)T^{2} \) |
| 61 | \( 1 + 29.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (52.3 - 90.6i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (3.50 + 33.3i)T + (-4.93e3 + 1.04e3i)T^{2} \) |
| 73 | \( 1 + (26.2 + 58.9i)T + (-3.56e3 + 3.96e3i)T^{2} \) |
| 79 | \( 1 + (48.1 - 108. i)T + (-4.17e3 - 4.63e3i)T^{2} \) |
| 83 | \( 1 + (-50.4 + 45.4i)T + (720. - 6.85e3i)T^{2} \) |
| 89 | \( 1 + (13.5 - 18.6i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-58.7 - 42.6i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−16.31437014746191898618107004934, −14.80559776869058418269659609744, −14.03129627186881816356941707684, −12.24601916146367168555743653108, −11.69540440085638738442661120088, −10.36628342480326949253593707007, −9.123415534868255660860882029045, −6.84352966717461628669787463753, −4.24861539463463410390958126305, −2.76259623764070345264677161784,
4.44112845307701971141351094255, 5.88910950067908177136269687809, 7.79977876271709613408209330112, 8.274841423420342249833644302269, 10.55954617549578546779688419229, 12.63578808450490959326406732423, 13.44186459316346069326717518422, 14.76648404452962568816179962133, 15.72042977596911897188422767946, 16.77600982399354475652619077368
