L(s) = 1 | + (−5.32 − 9.22i)5-s + (−3.32 + 5.76i)11-s + 75.9·13-s + (−52.1 + 90.3i)17-s + (−42.7 − 74.0i)19-s + (−34.3 − 59.4i)23-s + (5.73 − 9.93i)25-s − 87.7·29-s + (−31.3 + 54.3i)31-s + (−21.1 − 36.5i)37-s − 313.·41-s + 306.·43-s + (107. + 186. i)47-s + (262. − 454. i)53-s + 70.8·55-s + ⋯ |
L(s) = 1 | + (−0.476 − 0.825i)5-s + (−0.0911 + 0.157i)11-s + 1.62·13-s + (−0.743 + 1.28i)17-s + (−0.516 − 0.893i)19-s + (−0.311 − 0.539i)23-s + (0.0458 − 0.0794i)25-s − 0.562·29-s + (−0.181 + 0.314i)31-s + (−0.0937 − 0.162i)37-s − 1.19·41-s + 1.08·43-s + (0.333 + 0.578i)47-s + (0.680 − 1.17i)53-s + 0.173·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.900 - 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.09863784264\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09863784264\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (5.32 + 9.22i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (3.32 - 5.76i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 75.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + (52.1 - 90.3i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (42.7 + 74.0i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (34.3 + 59.4i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 87.7T + 2.43e4T^{2} \) |
| 31 | \( 1 + (31.3 - 54.3i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (21.1 + 36.5i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 313.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-107. - 186. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-262. + 454. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-180. + 312. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (400. + 693. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-20.1 + 34.8i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 298.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (258. - 447. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-611. - 1.05e3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (319. + 554. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.42e3T + 9.12e5T^{2} \) |
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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
−8.617153271515018770126795694993, −7.950796520902237604747548881532, −6.76041857578602834460564930224, −6.15666329622540342246554583229, −5.13031680826914635869016699673, −4.22282236832704798007039063073, −3.65376084041197839428697235252, −2.21214385364944218870295928480, −1.15068886676920039789260908770, −0.02245101264375865693927351174,
1.37781871424947191598673515857, 2.62898846559399317047977388266, 3.56110011198134418312394408148, 4.20902637735629333198886911036, 5.50210204097166610516347552036, 6.21799725187302246833454777658, 7.07252596982694952133508323296, 7.70697979809965914599126138287, 8.649976234298025235084226143441, 9.235030520311866813630402370809