| L(s) = 1 | + (−2.02 + 1.47i)2-s + (0.927 + 2.85i)3-s + (−0.539 + 1.65i)4-s + (−8.44 − 6.13i)5-s + (−6.07 − 4.41i)6-s + (−10.1 + 31.1i)7-s + (−7.53 − 23.1i)8-s + (−7.28 + 5.29i)9-s + 26.0·10-s + (12.6 + 34.2i)11-s − 5.23·12-s + (59.2 − 43.0i)13-s + (−25.3 − 77.9i)14-s + (9.67 − 29.7i)15-s + (38.0 + 27.6i)16-s + (44.7 + 32.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.715 + 0.519i)2-s + (0.178 + 0.549i)3-s + (−0.0673 + 0.207i)4-s + (−0.755 − 0.548i)5-s + (−0.413 − 0.300i)6-s + (−0.546 + 1.68i)7-s + (−0.332 − 1.02i)8-s + (−0.269 + 0.195i)9-s + 0.825·10-s + (0.347 + 0.937i)11-s − 0.125·12-s + (1.26 − 0.918i)13-s + (−0.483 − 1.48i)14-s + (0.166 − 0.512i)15-s + (0.594 + 0.431i)16-s + (0.638 + 0.463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.818 - 0.574i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.200172 + 0.633972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.200172 + 0.633972i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.927 - 2.85i)T \) |
| 11 | \( 1 + (-12.6 - 34.2i)T \) |
| good | 2 | \( 1 + (2.02 - 1.47i)T + (2.47 - 7.60i)T^{2} \) |
| 5 | \( 1 + (8.44 + 6.13i)T + (38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (10.1 - 31.1i)T + (-277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-59.2 + 43.0i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-44.7 - 32.4i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (-27.9 - 86.1i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + 91.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + (23.7 - 72.9i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (18.6 - 13.5i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (-38.8 + 119. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-43.2 - 133. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 146.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (68.4 + 210. i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-35.5 + 25.8i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (124. - 382. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-328. - 238. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 221.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-606. - 440. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (68.6 - 211. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-954. + 693. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (547. + 397. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-198. + 144. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−16.33959455843402284544383510091, −15.83266812529374383414912999704, −14.94271224283849943382729282251, −12.71950113025708209448395331620, −12.03626425539602529805538209256, −9.923602876588567181554180581360, −8.787084771916623661692272837911, −7.998353157073153946090564481197, −5.88661804657719469367261424182, −3.67324718881870581718367017412,
0.790914546188214919336574994937, 3.61890354833987843907564749361, 6.52766807652638573261451184614, 7.88257078388935275236840308086, 9.396922299462231645443732301537, 10.85929524456993367132121237783, 11.51038723931540822952772803516, 13.59623476476450836005786504684, 14.16992502237985723429875309371, 15.97298613756146101575015579511
