| L(s) = 1 | + (−3.53 + 0.740i)2-s + (2.71 + 1.46i)3-s + (8.27 − 3.62i)4-s + (7.69 + 0.637i)5-s + (−10.6 − 3.17i)6-s + (−11.5 + 5.63i)7-s + (−14.7 + 10.5i)8-s + (0.283 + 0.434i)9-s + (−27.6 + 3.44i)10-s + (−16.6 + 4.22i)11-s + (27.7 + 2.30i)12-s + (−0.0907 + 0.0768i)13-s + (36.5 − 28.4i)14-s + (19.9 + 13.0i)15-s + (19.9 − 21.6i)16-s + (−1.49 + 18.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.76 + 0.370i)2-s + (0.904 + 0.489i)3-s + (2.06 − 0.907i)4-s + (1.53 + 0.127i)5-s + (−1.77 − 0.529i)6-s + (−1.64 + 0.804i)7-s + (−1.84 + 1.31i)8-s + (0.0315 + 0.0482i)9-s + (−2.76 + 0.344i)10-s + (−1.51 + 0.384i)11-s + (2.31 + 0.191i)12-s + (−0.00697 + 0.00591i)13-s + (2.60 − 2.03i)14-s + (1.32 + 0.868i)15-s + (1.24 − 1.35i)16-s + (−0.0878 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.172924 + 0.647779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.172924 + 0.647779i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 229 | \( 1 + (-228. - 17.3i)T \) |
| good | 2 | \( 1 + (3.53 - 0.740i)T + (3.66 - 1.60i)T^{2} \) |
| 3 | \( 1 + (-2.71 - 1.46i)T + (4.92 + 7.53i)T^{2} \) |
| 5 | \( 1 + (-7.69 - 0.637i)T + (24.6 + 4.11i)T^{2} \) |
| 7 | \( 1 + (11.5 - 5.63i)T + (30.0 - 38.6i)T^{2} \) |
| 11 | \( 1 + (16.6 - 4.22i)T + (106. - 57.5i)T^{2} \) |
| 13 | \( 1 + (0.0907 - 0.0768i)T + (27.8 - 166. i)T^{2} \) |
| 17 | \( 1 + (1.49 - 18.0i)T + (-285. - 47.5i)T^{2} \) |
| 19 | \( 1 + (-1.74 - 21.0i)T + (-356. + 59.4i)T^{2} \) |
| 23 | \( 1 + (10.2 + 1.28i)T + (512. + 129. i)T^{2} \) |
| 29 | \( 1 + (3.24 + 6.63i)T + (-516. + 663. i)T^{2} \) |
| 31 | \( 1 + (14.4 - 24.1i)T + (-457. - 845. i)T^{2} \) |
| 37 | \( 1 + (-49.1 - 53.4i)T + (-113. + 1.36e3i)T^{2} \) |
| 41 | \( 1 + (8.36 + 39.8i)T + (-1.53e3 + 675. i)T^{2} \) |
| 43 | \( 1 + (-55.4 - 60.2i)T + (-152. + 1.84e3i)T^{2} \) |
| 47 | \( 1 + (-26.6 - 5.58i)T + (2.02e3 + 887. i)T^{2} \) |
| 53 | \( 1 + (17.5 + 9.49i)T + (1.53e3 + 2.35e3i)T^{2} \) |
| 59 | \( 1 + (36.4 - 1.50i)T + (3.46e3 - 287. i)T^{2} \) |
| 61 | \( 1 + (20.3 + 31.1i)T + (-1.49e3 + 3.40e3i)T^{2} \) |
| 67 | \( 1 + (-18.9 - 3.98i)T + (4.11e3 + 1.80e3i)T^{2} \) |
| 71 | \( 1 + (34.3 + 8.68i)T + (4.43e3 + 2.39e3i)T^{2} \) |
| 73 | \( 1 + (22.4 + 31.4i)T + (-1.73e3 + 5.04e3i)T^{2} \) |
| 79 | \( 1 + (-3.24 + 6.63i)T + (-3.83e3 - 4.92e3i)T^{2} \) |
| 83 | \( 1 + (-20.4 + 22.2i)T + (-568. - 6.86e3i)T^{2} \) |
| 89 | \( 1 + (1.41 + 1.41i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-5.56 - 33.3i)T + (-8.89e3 + 3.05e3i)T^{2} \) |
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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
−12.44814185536298668656798823813, −10.52886504367454868088863191791, −9.986437835159422314367679168655, −9.534988862946318635249779149466, −8.776525489718091117856503243993, −7.81364122541955866545367102127, −6.28926092809395578704473938838, −5.87760890962836904941661532856, −2.97023181820988473333914945242, −2.11139071940662160787925019451,
0.51358615173290550106806142200, 2.36911422567584247869089960945, 2.86895728306810249765113614237, 5.81604514057593132287718280417, 7.05774991120569840479420880493, 7.69249754245912474899855324125, 9.071909038370759579120178731494, 9.426551343532948480358872837789, 10.27593198830517222526611719120, 10.98271950478146301441862580336
