| L(s) = 1 | + (1.05 − 4.61i)2-s + (0.0821 − 0.0762i)3-s + (−12.9 − 6.24i)4-s + (−9.16 − 1.38i)5-s + (−0.265 − 0.459i)6-s + (13.3 − 23.1i)7-s + (−18.8 + 23.6i)8-s + (−2.01 + 26.9i)9-s + (−16.0 + 40.8i)10-s + (46.6 − 22.4i)11-s + (−1.54 + 0.475i)12-s + (7.16 + 18.2i)13-s + (−92.6 − 85.9i)14-s + (−0.858 + 0.585i)15-s + (17.5 + 21.9i)16-s + (25.7 − 3.88i)17-s + ⋯ |
| L(s) = 1 | + (0.372 − 1.63i)2-s + (0.0158 − 0.0146i)3-s + (−1.62 − 0.780i)4-s + (−0.819 − 0.123i)5-s + (−0.0180 − 0.0312i)6-s + (0.720 − 1.24i)7-s + (−0.834 + 1.04i)8-s + (−0.0746 + 0.996i)9-s + (−0.506 + 1.29i)10-s + (1.27 − 0.615i)11-s + (−0.0370 + 0.0114i)12-s + (0.152 + 0.389i)13-s + (−1.76 − 1.64i)14-s + (−0.0147 + 0.0100i)15-s + (0.273 + 0.343i)16-s + (0.367 − 0.0553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.312018 - 1.36402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.312018 - 1.36402i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-91.6 - 266. i)T \) |
| good | 2 | \( 1 + (-1.05 + 4.61i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (-0.0821 + 0.0762i)T + (2.01 - 26.9i)T^{2} \) |
| 5 | \( 1 + (9.16 + 1.38i)T + (119. + 36.8i)T^{2} \) |
| 7 | \( 1 + (-13.3 + 23.1i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-46.6 + 22.4i)T + (829. - 1.04e3i)T^{2} \) |
| 13 | \( 1 + (-7.16 - 18.2i)T + (-1.61e3 + 1.49e3i)T^{2} \) |
| 17 | \( 1 + (-25.7 + 3.88i)T + (4.69e3 - 1.44e3i)T^{2} \) |
| 19 | \( 1 + (2.15 + 28.6i)T + (-6.78e3 + 1.02e3i)T^{2} \) |
| 23 | \( 1 + (-101. - 69.2i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (-152. - 141. i)T + (1.82e3 + 2.43e4i)T^{2} \) |
| 31 | \( 1 + (251. - 77.7i)T + (2.46e4 - 1.67e4i)T^{2} \) |
| 37 | \( 1 + (36.8 + 63.8i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-37.6 + 164. i)T + (-6.20e4 - 2.99e4i)T^{2} \) |
| 47 | \( 1 + (-447. - 215. i)T + (6.47e4 + 8.11e4i)T^{2} \) |
| 53 | \( 1 + (197. - 502. i)T + (-1.09e5 - 1.01e5i)T^{2} \) |
| 59 | \( 1 + (481. + 604. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-150. - 46.4i)T + (1.87e5 + 1.27e5i)T^{2} \) |
| 67 | \( 1 + (-11.8 - 158. i)T + (-2.97e5 + 4.48e4i)T^{2} \) |
| 71 | \( 1 + (-796. + 543. i)T + (1.30e5 - 3.33e5i)T^{2} \) |
| 73 | \( 1 + (341. + 869. i)T + (-2.85e5 + 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-130. + 225. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (463. - 430. i)T + (4.27e4 - 5.70e5i)T^{2} \) |
| 89 | \( 1 + (-275. + 255. i)T + (5.26e4 - 7.02e5i)T^{2} \) |
| 97 | \( 1 + (-891. + 429. i)T + (5.69e5 - 7.13e5i)T^{2} \) |
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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
−14.26644822353308066030861864123, −13.75515649473441221863449870425, −12.34780050404110458424429107958, −11.16390852930208849461071762751, −10.81962367286215529529469071027, −9.073634830924813148843392419742, −7.47361027852601353743196283532, −4.70593019091015788659976322908, −3.62739965189245170599214120558, −1.24798980462878822709838682821,
4.06755542899030581705861978611, 5.65500072388307057775122382388, 6.90852726634691497560095904500, 8.238205885357219813784462264783, 9.182125838936056902539542114030, 11.64089192282719536590479214592, 12.50844021529217395626253378007, 14.34926782348257943531511523838, 15.03836742837622385510136403163, 15.50774865994047226816893091851
