| L(s) = 1 | + (−6.08 − 4.84i)2-s + (2.22 + 14.7i)3-s + (9.90 + 43.3i)4-s + (−17.9 − 26.3i)5-s + (58.1 − 100. i)6-s + (−7.53 + 4.35i)7-s + (96.1 − 199. i)8-s + (−136. + 41.9i)9-s + (−18.5 + 247. i)10-s + (14.4 − 63.5i)11-s + (−619. + 243. i)12-s + (−12.2 − 163. i)13-s + (66.9 + 10.0i)14-s + (349. − 324. i)15-s + (−911. + 439. i)16-s + (−199. − 135. i)17-s + ⋯ |
| L(s) = 1 | + (−1.52 − 1.21i)2-s + (0.247 + 1.64i)3-s + (0.618 + 2.71i)4-s + (−0.718 − 1.05i)5-s + (1.61 − 2.79i)6-s + (−0.153 + 0.0887i)7-s + (1.50 − 3.12i)8-s + (−1.68 + 0.518i)9-s + (−0.185 + 2.47i)10-s + (0.119 − 0.524i)11-s + (−4.29 + 1.68i)12-s + (−0.0722 − 0.964i)13-s + (0.341 + 0.0514i)14-s + (1.55 − 1.44i)15-s + (−3.56 + 1.71i)16-s + (−0.688 − 0.469i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0716142 - 0.250422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0716142 - 0.250422i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.46e3 - 1.12e3i)T \) |
| good | 2 | \( 1 + (6.08 + 4.84i)T + (3.56 + 15.5i)T^{2} \) |
| 3 | \( 1 + (-2.22 - 14.7i)T + (-77.4 + 23.8i)T^{2} \) |
| 5 | \( 1 + (17.9 + 26.3i)T + (-228. + 581. i)T^{2} \) |
| 7 | \( 1 + (7.53 - 4.35i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-14.4 + 63.5i)T + (-1.31e4 - 6.35e3i)T^{2} \) |
| 13 | \( 1 + (12.2 + 163. i)T + (-2.82e4 + 4.25e3i)T^{2} \) |
| 17 | \( 1 + (199. + 135. i)T + (3.05e4 + 7.77e4i)T^{2} \) |
| 19 | \( 1 + (-53.5 + 173. i)T + (-1.07e5 - 7.34e4i)T^{2} \) |
| 23 | \( 1 + (242. + 224. i)T + (2.09e4 + 2.79e5i)T^{2} \) |
| 29 | \( 1 + (34.6 - 230. i)T + (-6.75e5 - 2.08e5i)T^{2} \) |
| 31 | \( 1 + (384. + 979. i)T + (-6.76e5 + 6.28e5i)T^{2} \) |
| 37 | \( 1 + (-149. - 86.4i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (1.76e3 - 2.20e3i)T + (-6.28e5 - 2.75e6i)T^{2} \) |
| 47 | \( 1 + (578. + 2.53e3i)T + (-4.39e6 + 2.11e6i)T^{2} \) |
| 53 | \( 1 + (-240. + 3.21e3i)T + (-7.80e6 - 1.17e6i)T^{2} \) |
| 59 | \( 1 + (3.76e3 - 1.81e3i)T + (7.55e6 - 9.47e6i)T^{2} \) |
| 61 | \( 1 + (1.43e3 + 562. i)T + (1.01e7 + 9.41e6i)T^{2} \) |
| 67 | \( 1 + (5.36e3 + 1.65e3i)T + (1.66e7 + 1.13e7i)T^{2} \) |
| 71 | \( 1 + (-2.39e3 - 2.57e3i)T + (-1.89e6 + 2.53e7i)T^{2} \) |
| 73 | \( 1 + (2.63e3 - 197. i)T + (2.80e7 - 4.23e6i)T^{2} \) |
| 79 | \( 1 + (-248. - 429. i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-4.16e3 + 628. i)T + (4.53e7 - 1.39e7i)T^{2} \) |
| 89 | \( 1 + (-802. - 5.32e3i)T + (-5.99e7 + 1.84e7i)T^{2} \) |
| 97 | \( 1 + (-3.98e3 + 1.74e4i)T + (-7.97e7 - 3.84e7i)T^{2} \) |
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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
−15.41525831354133526836060806953, −13.04998471063218888786877417371, −11.73583576495321713330704856111, −10.84198659868834639420246223015, −9.728360738070213812860839262540, −8.883764438972570707832423574417, −8.060403464592713789260021920580, −4.46436066796943498644023119678, −3.13995654860650132983580438829, −0.26097513999164333071502313414,
1.81556440752245214402412982141, 6.29981366244010771133490033094, 7.08341635908561508619960765743, 7.74569963885810005638908714805, 9.007207506206262755655132468522, 10.65232035146132381319418555276, 11.91204822275483136626173328725, 13.87875767585929039162807522134, 14.69253339733819212581235719100, 15.76693252571731449608949021230
