| L(s) = 1 | + (−8.43 + 6.12i)2-s + (−8.34 − 25.6i)3-s + (−5.95 + 18.3i)4-s + (−186. − 135. i)5-s + (227. + 165. i)6-s + (−43.6 + 134. i)7-s + (−474. − 1.46e3i)8-s + (−589. + 428. i)9-s + 2.40e3·10-s + (4.40e3 − 283. i)11-s + 520.·12-s + (−2.40e3 + 1.74e3i)13-s + (−455. − 1.40e3i)14-s + (−1.92e3 + 5.91e3i)15-s + (1.09e4 + 7.96e3i)16-s + (3.21e4 + 2.33e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.745 + 0.541i)2-s + (−0.178 − 0.549i)3-s + (−0.0465 + 0.143i)4-s + (−0.667 − 0.484i)5-s + (0.430 + 0.312i)6-s + (−0.0481 + 0.148i)7-s + (−0.327 − 1.00i)8-s + (−0.269 + 0.195i)9-s + 0.759·10-s + (0.997 − 0.0642i)11-s + 0.0869·12-s + (−0.303 + 0.220i)13-s + (−0.0443 − 0.136i)14-s + (−0.147 + 0.452i)15-s + (0.668 + 0.485i)16-s + (1.58 + 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.784 - 0.620i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.836827 + 0.290775i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.836827 + 0.290775i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (8.34 + 25.6i)T \) |
| 11 | \( 1 + (-4.40e3 + 283. i)T \) |
| good | 2 | \( 1 + (8.43 - 6.12i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (186. + 135. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (43.6 - 134. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (2.40e3 - 1.74e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-3.21e4 - 2.33e4i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-1.13e4 - 3.48e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 2.62e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-6.22e4 + 1.91e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-3.44e4 + 2.49e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (9.33e4 - 2.87e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (1.66e5 + 5.12e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 2.82e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-2.25e5 - 6.94e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (3.67e5 - 2.67e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-3.67e5 + 1.13e6i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-3.53e5 - 2.57e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 2.93e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-6.39e5 - 4.64e5i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (2.28e4 - 7.02e4i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (4.91e6 - 3.56e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-1.60e6 - 1.16e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 - 7.69e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-6.55e5 + 4.76e5i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−15.64180543239583433139036311062, −14.21415427628370054120270477109, −12.48695731104844452162514872622, −11.96275552501840528469264873577, −9.886035104633389158516261114071, −8.459365686506401030512647067468, −7.63471223017052020664949569700, −6.14377861475314490126937269692, −3.84124655619635044761005731406, −0.982538838645283212520506203394,
0.78811497474203453907951148270, 3.18495491223589133247339058214, 5.15955704018236284775399060892, 7.21641016370376897563999556751, 8.961580837261838753115163843807, 9.973724311339665294369219668180, 11.15387624531102627671405641384, 11.97520713103952052691211634317, 14.15517124961832857716224301743, 15.01334153615766802315032286004
