L(s) = 1 | + (−12.2 − 9.62i)3-s + (−14.0 + 24.3i)5-s + (75.7 + 131. i)7-s + (57.8 + 236. i)9-s + (138. + 240. i)11-s + (−291. + 505. i)13-s + (407. − 163. i)15-s − 1.61e3·17-s + 1.36e3·19-s + (333. − 2.33e3i)21-s + (−428. + 741. i)23-s + (1.16e3 + 2.02e3i)25-s + (1.56e3 − 3.45e3i)27-s + (−4.26e3 − 7.39e3i)29-s + (−1.46e3 + 2.54e3i)31-s + ⋯ |
L(s) = 1 | + (−0.786 − 0.617i)3-s + (−0.251 + 0.435i)5-s + (0.583 + 1.01i)7-s + (0.238 + 0.971i)9-s + (0.346 + 0.599i)11-s + (−0.479 + 0.829i)13-s + (0.467 − 0.187i)15-s − 1.35·17-s + 0.869·19-s + (0.164 − 1.15i)21-s + (−0.168 + 0.292i)23-s + (0.373 + 0.646i)25-s + (0.412 − 0.911i)27-s + (−0.942 − 1.63i)29-s + (−0.274 + 0.475i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.108 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.696986 + 0.625051i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.696986 + 0.625051i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (12.2 + 9.62i)T \) |
good | 5 | \( 1 + (14.0 - 24.3i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-75.7 - 131. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-138. - 240. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (291. - 505. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 1.61e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.36e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (428. - 741. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (4.26e3 + 7.39e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (1.46e3 - 2.54e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 4.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (9.44e3 - 1.63e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.01e4 - 1.75e4i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-147. - 256. i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 - 3.03e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-8.61e3 + 1.49e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.28e4 + 2.22e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.31e4 + 2.27e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 7.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.49e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (4.96e4 + 8.59e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (2.50e4 + 4.33e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 1.36e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-3.33e4 - 5.77e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−15.68533978810857972349864894534, −14.60716578014636527424232736109, −13.14819520722082835221845750840, −11.79728642845994429133629043182, −11.28696196984408255411580676379, −9.408832080248100072635627918830, −7.68079570728919165771519674565, −6.40872402117985270357826622893, −4.82476249176757688738235428179, −2.02586065988618080662748488835,
0.61280684167769166418064946995, 3.99534998016041612546391021968, 5.31591519936661240673279455411, 7.13551762706558436450904501546, 8.847819119115101233458536798449, 10.42367651389948802072662403989, 11.27805814222494630618484829898, 12.58037362207655876212564221989, 14.03740583238221792034318724595, 15.36837567297588219920101949888