| L(s) = 1 | + (0.747 + 3.27i)2-s + (−6.89 − 6.39i)3-s + (−2.96 + 1.42i)4-s + (−13.7 + 2.06i)5-s + (15.8 − 27.3i)6-s + (−15.9 − 27.5i)7-s + (9.87 + 12.3i)8-s + (4.59 + 61.3i)9-s + (−17.0 − 43.4i)10-s + (4.43 + 2.13i)11-s + (29.5 + 9.11i)12-s + (4.71 − 12.0i)13-s + (78.3 − 72.7i)14-s + (107. + 73.6i)15-s + (−49.5 + 62.1i)16-s + (−86.4 − 13.0i)17-s + ⋯ |
| L(s) = 1 | + (0.264 + 1.15i)2-s + (−1.32 − 1.23i)3-s + (−0.370 + 0.178i)4-s + (−1.22 + 0.185i)5-s + (1.07 − 1.86i)6-s + (−0.859 − 1.48i)7-s + (0.436 + 0.547i)8-s + (0.170 + 2.27i)9-s + (−0.539 − 1.37i)10-s + (0.121 + 0.0585i)11-s + (0.710 + 0.219i)12-s + (0.100 − 0.256i)13-s + (1.49 − 1.38i)14-s + (1.85 + 1.26i)15-s + (−0.774 + 0.971i)16-s + (−1.23 − 0.185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.175459 - 0.284377i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.175459 - 0.284377i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (265. + 95.7i)T \) |
| good | 2 | \( 1 + (-0.747 - 3.27i)T + (-7.20 + 3.47i)T^{2} \) |
| 3 | \( 1 + (6.89 + 6.39i)T + (2.01 + 26.9i)T^{2} \) |
| 5 | \( 1 + (13.7 - 2.06i)T + (119. - 36.8i)T^{2} \) |
| 7 | \( 1 + (15.9 + 27.5i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-4.43 - 2.13i)T + (829. + 1.04e3i)T^{2} \) |
| 13 | \( 1 + (-4.71 + 12.0i)T + (-1.61e3 - 1.49e3i)T^{2} \) |
| 17 | \( 1 + (86.4 + 13.0i)T + (4.69e3 + 1.44e3i)T^{2} \) |
| 19 | \( 1 + (-5.50 + 73.4i)T + (-6.78e3 - 1.02e3i)T^{2} \) |
| 23 | \( 1 + (-15.6 + 10.6i)T + (4.44e3 - 1.13e4i)T^{2} \) |
| 29 | \( 1 + (-126. + 117. i)T + (1.82e3 - 2.43e4i)T^{2} \) |
| 31 | \( 1 + (21.4 + 6.61i)T + (2.46e4 + 1.67e4i)T^{2} \) |
| 37 | \( 1 + (-93.5 + 161. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-23.3 - 102. i)T + (-6.20e4 + 2.99e4i)T^{2} \) |
| 47 | \( 1 + (190. - 91.5i)T + (6.47e4 - 8.11e4i)T^{2} \) |
| 53 | \( 1 + (104. + 266. i)T + (-1.09e5 + 1.01e5i)T^{2} \) |
| 59 | \( 1 + (6.98 - 8.75i)T + (-4.57e4 - 2.00e5i)T^{2} \) |
| 61 | \( 1 + (528. - 162. i)T + (1.87e5 - 1.27e5i)T^{2} \) |
| 67 | \( 1 + (-37.3 + 497. i)T + (-2.97e5 - 4.48e4i)T^{2} \) |
| 71 | \( 1 + (231. + 157. i)T + (1.30e5 + 3.33e5i)T^{2} \) |
| 73 | \( 1 + (34.3 - 87.5i)T + (-2.85e5 - 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-274. - 475. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-750. - 696. i)T + (4.27e4 + 5.70e5i)T^{2} \) |
| 89 | \( 1 + (816. + 757. i)T + (5.26e4 + 7.02e5i)T^{2} \) |
| 97 | \( 1 + (-229. - 110. i)T + (5.69e5 + 7.13e5i)T^{2} \) |
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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
−15.49073920481067672115405867722, −13.73990499348692486054123297722, −12.98374131876948941549549412328, −11.51347110578653667124458627865, −10.76205595434465907362637290868, −7.85645978910847186893955228195, −7.02650678123708439236221416766, −6.43020722783451027475446849135, −4.56915006073144492266965367056, −0.28177512994077662627678286303,
3.44248647461044298167819674586, 4.68345409666099871256759106575, 6.35654445135198111153367360692, 8.991116865956692088921737274482, 10.21028326382140928221200857837, 11.38540014889217929295855090497, 11.91918968149888577705282360460, 12.69090072984922604787037189749, 15.23373327209334533478557904670, 15.87668854795800485190033183872
