L(s) = 1 | + (−0.142 − 0.989i)2-s + (1.30 − 2.86i)3-s + (−0.959 + 0.281i)4-s + (0.654 − 0.755i)5-s + (−3.02 − 0.887i)6-s + (−0.493 + 0.316i)7-s + (0.415 + 0.909i)8-s + (−4.54 − 5.23i)9-s + (−0.841 − 0.540i)10-s + (−0.242 + 1.68i)11-s + (−0.448 + 3.11i)12-s + (5.25 + 3.37i)13-s + (0.383 + 0.442i)14-s + (−1.30 − 2.86i)15-s + (0.841 − 0.540i)16-s + (−2.14 − 0.629i)17-s + ⋯ |
L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.755 − 1.65i)3-s + (−0.479 + 0.140i)4-s + (0.292 − 0.337i)5-s + (−1.23 − 0.362i)6-s + (−0.186 + 0.119i)7-s + (0.146 + 0.321i)8-s + (−1.51 − 1.74i)9-s + (−0.266 − 0.170i)10-s + (−0.0730 + 0.507i)11-s + (−0.129 + 0.900i)12-s + (1.45 + 0.936i)13-s + (0.102 + 0.118i)14-s + (−0.338 − 0.740i)15-s + (0.210 − 0.135i)16-s + (−0.519 − 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.772 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.478829 - 1.33590i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.478829 - 1.33590i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (1.29 - 4.61i)T \) |
good | 3 | \( 1 + (-1.30 + 2.86i)T + (-1.96 - 2.26i)T^{2} \) |
| 7 | \( 1 + (0.493 - 0.316i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.242 - 1.68i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-5.25 - 3.37i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (2.14 + 0.629i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.47 + 0.432i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.262 - 0.0771i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (3.69 + 8.09i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-5.54 - 6.39i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.02 + 2.33i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-3.08 + 6.75i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 + (-8.65 + 5.56i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (5.47 + 3.51i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.86 - 10.6i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.87 - 13.0i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.13 + 7.86i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (1.37 - 0.404i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-9.95 - 6.39i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (2.84 + 3.28i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (4.46 - 9.77i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (2.54 - 2.93i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.91835236135837599388146321825, −11.27495789523236440618142129795, −9.582703839540458727822919290450, −8.889593727854218318721138297128, −7.985737569606796333956670977649, −6.91057026424914019357368863857, −5.85500961133866418921823645270, −3.88557723802176252282274929653, −2.40237909271653584767154534673, −1.34798592925105047147057560568,
3.06048292033531447780619307370, 4.01962342838275411488109641280, 5.29730185043216789137664766675, 6.30607228511383156997884454940, 7.998940108430077263234771144142, 8.713384880752405802461334390731, 9.542232593836261915656877137069, 10.57946042230589324031767735295, 10.97785831736565610681286303617, 12.98027864824461643550529965157