| L(s) = 1 | + (−0.873 + 1.09i)2-s + (4.80 + 6.02i)3-s + (1.34 + 5.88i)4-s + (−14.6 − 7.06i)5-s − 10.8·6-s + 16.8·7-s + (−17.7 − 8.53i)8-s + (−7.22 + 31.6i)9-s + (20.5 − 9.90i)10-s + (−3.70 + 16.2i)11-s + (−29.0 + 36.3i)12-s + (72.4 + 34.8i)13-s + (−14.7 + 18.4i)14-s + (−27.9 − 122. i)15-s + (−18.6 + 8.99i)16-s + (77.8 − 37.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.308 + 0.387i)2-s + (0.925 + 1.16i)3-s + (0.167 + 0.735i)4-s + (−1.31 − 0.632i)5-s − 0.735·6-s + 0.908·7-s + (−0.783 − 0.377i)8-s + (−0.267 + 1.17i)9-s + (0.650 − 0.313i)10-s + (−0.101 + 0.444i)11-s + (−0.698 + 0.875i)12-s + (1.54 + 0.744i)13-s + (−0.280 + 0.351i)14-s + (−0.481 − 2.10i)15-s + (−0.291 + 0.140i)16-s + (1.11 − 0.535i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.785563 + 1.09245i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.785563 + 1.09245i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (48.3 + 277. i)T \) |
| good | 2 | \( 1 + (0.873 - 1.09i)T + (-1.78 - 7.79i)T^{2} \) |
| 3 | \( 1 + (-4.80 - 6.02i)T + (-6.00 + 26.3i)T^{2} \) |
| 5 | \( 1 + (14.6 + 7.06i)T + (77.9 + 97.7i)T^{2} \) |
| 7 | \( 1 - 16.8T + 343T^{2} \) |
| 11 | \( 1 + (3.70 - 16.2i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (-72.4 - 34.8i)T + (1.36e3 + 1.71e3i)T^{2} \) |
| 17 | \( 1 + (-77.8 + 37.5i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 + (29.9 + 131. i)T + (-6.17e3 + 2.97e3i)T^{2} \) |
| 23 | \( 1 + (20.6 - 90.4i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (-30.9 + 38.7i)T + (-5.42e3 - 2.37e4i)T^{2} \) |
| 31 | \( 1 + (37.2 - 46.7i)T + (-6.62e3 - 2.90e4i)T^{2} \) |
| 37 | \( 1 + 128.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-152. + 191. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 47 | \( 1 + (17.3 + 75.8i)T + (-9.35e4 + 4.50e4i)T^{2} \) |
| 53 | \( 1 + (400. - 192. i)T + (9.28e4 - 1.16e5i)T^{2} \) |
| 59 | \( 1 + (485. - 233. i)T + (1.28e5 - 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-391. - 490. i)T + (-5.05e4 + 2.21e5i)T^{2} \) |
| 67 | \( 1 + (121. + 533. i)T + (-2.70e5 + 1.30e5i)T^{2} \) |
| 71 | \( 1 + (50.5 + 221. i)T + (-3.22e5 + 1.55e5i)T^{2} \) |
| 73 | \( 1 + (663. + 319. i)T + (2.42e5 + 3.04e5i)T^{2} \) |
| 79 | \( 1 - 440.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (53.2 + 66.8i)T + (-1.27e5 + 5.57e5i)T^{2} \) |
| 89 | \( 1 + (-550. - 689. i)T + (-1.56e5 + 6.87e5i)T^{2} \) |
| 97 | \( 1 + (-1.55 + 6.81i)T + (-8.22e5 - 3.95e5i)T^{2} \) |
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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
−15.75447394798871305381480793450, −15.19682201741115527565994828292, −13.75505299254438800294632178536, −12.07815917985841002209479226969, −11.09740532559193747083569265636, −9.126620141402675016636503421263, −8.473891500171184668858010294833, −7.47208858578504970145808085037, −4.56044769699492311988297520121, −3.52208445989155807510196968642,
1.35735315134370808907838088324, 3.33847979965803905854288283618, 6.18780171944983480591057405954, 7.949370662968995439554438782994, 8.316987753672268010605447659385, 10.53451167195218689305253079937, 11.44923836777170196962662546800, 12.66293265127770133909587885425, 14.32792221524802306009701032240, 14.67429514421890343083089626291
