| L(s) = 1 | + (−2.29 + 2.87i)2-s + (−5.14 − 6.45i)3-s + (−1.23 − 5.42i)4-s + (11.0 + 5.33i)5-s + 30.4·6-s + 28.5·7-s + (−8.06 − 3.88i)8-s + (−9.15 + 40.1i)9-s + (−40.7 + 19.6i)10-s + (12.4 − 54.4i)11-s + (−28.6 + 35.9i)12-s + (63.9 + 30.7i)13-s + (−65.5 + 82.2i)14-s + (−22.5 − 98.9i)15-s + (69.8 − 33.6i)16-s + (−18.8 + 9.07i)17-s + ⋯ |
| L(s) = 1 | + (−0.811 + 1.01i)2-s + (−0.990 − 1.24i)3-s + (−0.154 − 0.678i)4-s + (0.990 + 0.477i)5-s + 2.06·6-s + 1.54·7-s + (−0.356 − 0.171i)8-s + (−0.339 + 1.48i)9-s + (−1.29 + 0.621i)10-s + (0.340 − 1.49i)11-s + (−0.689 + 0.864i)12-s + (1.36 + 0.656i)13-s + (−1.25 + 1.57i)14-s + (−0.388 − 1.70i)15-s + (1.09 − 0.525i)16-s + (−0.268 + 0.129i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.850188 + 0.120461i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.850188 + 0.120461i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-279. + 36.5i)T \) |
| good | 2 | \( 1 + (2.29 - 2.87i)T + (-1.78 - 7.79i)T^{2} \) |
| 3 | \( 1 + (5.14 + 6.45i)T + (-6.00 + 26.3i)T^{2} \) |
| 5 | \( 1 + (-11.0 - 5.33i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 - 28.5T + 343T^{2} \) |
| 11 | \( 1 + (-12.4 + 54.4i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-63.9 - 30.7i)T + (1.36e3 + 1.71e3i)T^{2} \) |
| 17 | \( 1 + (18.8 - 9.07i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + (5.65 + 24.7i)T + (-6.17e3 + 2.97e3i)T^{2} \) |
| 23 | \( 1 + (23.0 - 101. i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (-58.6 + 73.5i)T + (-5.42e3 - 2.37e4i)T^{2} \) |
| 31 | \( 1 + (-10.0 + 12.5i)T + (-6.62e3 - 2.90e4i)T^{2} \) |
| 37 | \( 1 + 23.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + (47.1 - 59.0i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 47 | \( 1 + (-1.23 - 5.42i)T + (-9.35e4 + 4.50e4i)T^{2} \) |
| 53 | \( 1 + (560. - 269. i)T + (9.28e4 - 1.16e5i)T^{2} \) |
| 59 | \( 1 + (170. - 82.2i)T + (1.28e5 - 1.60e5i)T^{2} \) |
| 61 | \( 1 + (231. + 290. i)T + (-5.05e4 + 2.21e5i)T^{2} \) |
| 67 | \( 1 + (-138. - 605. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (143. + 628. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (440. + 212. i)T + (2.42e5 + 3.04e5i)T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (219. + 275. i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (267. + 335. i)T + (-1.56e5 + 6.87e5i)T^{2} \) |
| 97 | \( 1 + (263. - 1.15e3i)T + (-8.22e5 - 3.95e5i)T^{2} \) |
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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
−15.93146918664607977005387773052, −14.22589214846418679250968916915, −13.50631753846452856418195011304, −11.68057780555049754288649460235, −10.93169231699043543289603556307, −8.842439038312615301390053527107, −7.77128070070474023420410030355, −6.38016006126985579825538448751, −5.81970326778998665350990587044, −1.33450872679482291070369605862,
1.55773980287843326111863393028, 4.54503652522711654279903993119, 5.76048101352273142798389448658, 8.574704454282960195025083288300, 9.711031577607863419917992032457, 10.57248429459648341659212897162, 11.32286192997998952217174914040, 12.49994708033666232286227899822, 14.44756619729322244014520896193, 15.57321214374340199792451438829
