L(s) = 1 | + (−8.67 − 12.9i)3-s + (13.1 + 22.7i)5-s + (31.6 − 54.7i)7-s + (−92.4 + 224. i)9-s + (49.1 − 85.0i)11-s + (369. + 639. i)13-s + (181. − 368. i)15-s + 250.·17-s − 1.10e3·19-s + (−983. + 65.6i)21-s + (2.20e3 + 3.81e3i)23-s + (1.21e3 − 2.10e3i)25-s + (3.71e3 − 752. i)27-s + (3.94e3 − 6.82e3i)29-s + (−2.30e3 − 3.99e3i)31-s + ⋯ |
L(s) = 1 | + (−0.556 − 0.830i)3-s + (0.235 + 0.407i)5-s + (0.243 − 0.422i)7-s + (−0.380 + 0.924i)9-s + (0.122 − 0.211i)11-s + (0.605 + 1.04i)13-s + (0.207 − 0.422i)15-s + 0.209·17-s − 0.700·19-s + (−0.486 + 0.0325i)21-s + (0.869 + 1.50i)23-s + (0.389 − 0.674i)25-s + (0.980 − 0.198i)27-s + (0.870 − 1.50i)29-s + (−0.430 − 0.746i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.844 + 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.694898003\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.694898003\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (8.67 + 12.9i)T \) |
good | 5 | \( 1 + (-13.1 - 22.7i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-31.6 + 54.7i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-49.1 + 85.0i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-369. - 639. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 250.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.10e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-2.20e3 - 3.81e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-3.94e3 + 6.82e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.30e3 + 3.99e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.18e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (5.04e3 + 8.73e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-3.51e3 + 6.09e3i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-7.45e3 + 1.29e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 2.24e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-5.40e3 - 9.36e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-594. + 1.02e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.95e4 - 5.12e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (1.86e4 - 3.23e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-6.04e4 + 1.04e5i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 9.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (5.33e4 - 9.24e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−11.91096238496291458157612769961, −11.25777713272910346887093116389, −10.28215463483922571521383411601, −8.856611149868042333264072497579, −7.62667837010761860943649941408, −6.67167941703508890524843057031, −5.73890694151454468209696013094, −4.17180262203062433352223088707, −2.29327695348439657852659631307, −0.900860932988173695075717566957,
0.945023024305247521576134489162, 3.04636596667796510992060096348, 4.59485639449961460097531465108, 5.48044817795776089091754511701, 6.62497491582505384619163453512, 8.398030271362546457295197652604, 9.150295039024062656541335267344, 10.41563121624029480883079857197, 11.03043690140155687103227119823, 12.33229534106697975269341288618