| L(s) = 1 | + (−1.12 + 0.861i)2-s + (1.64 − 0.528i)3-s + (0.516 − 1.93i)4-s + (1.99 − 1.00i)5-s + (−1.39 + 2.01i)6-s − 3.00·7-s + (1.08 + 2.61i)8-s + (2.44 − 1.74i)9-s + (−1.37 + 2.84i)10-s + (3.97 + 2.88i)11-s + (−0.168 − 3.45i)12-s + (−1.00 − 1.37i)13-s + (3.37 − 2.58i)14-s + (2.76 − 2.71i)15-s + (−3.46 − 1.99i)16-s + (1.60 − 4.94i)17-s + ⋯ |
| L(s) = 1 | + (−0.793 + 0.608i)2-s + (0.952 − 0.305i)3-s + (0.258 − 0.966i)4-s + (0.892 − 0.450i)5-s + (−0.569 + 0.821i)6-s − 1.13·7-s + (0.383 + 0.923i)8-s + (0.813 − 0.581i)9-s + (−0.434 + 0.900i)10-s + (1.19 + 0.870i)11-s + (−0.0486 − 0.998i)12-s + (−0.277 − 0.381i)13-s + (0.901 − 0.692i)14-s + (0.712 − 0.701i)15-s + (−0.866 − 0.499i)16-s + (0.389 − 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32471 - 0.0564092i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32471 - 0.0564092i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.12 - 0.861i)T \) |
| 3 | \( 1 + (-1.64 + 0.528i)T \) |
| 5 | \( 1 + (-1.99 + 1.00i)T \) |
| good | 7 | \( 1 + 3.00T + 7T^{2} \) |
| 11 | \( 1 + (-3.97 - 2.88i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (1.00 + 1.37i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.60 + 4.94i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.378 + 0.122i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.680 - 0.936i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (4.13 - 1.34i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.06 - 1.64i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.06 - 2.84i)T + (-11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.11 - 8.42i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 10.9T + 43T^{2} \) |
| 47 | \( 1 + (10.9 - 3.56i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.93 - 9.02i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.303 + 0.220i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (6.49 + 4.71i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.185 + 0.570i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.83 + 8.71i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (6.41 - 8.83i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-2.30 + 0.749i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.583 - 0.189i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (2.36 - 3.24i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.41 - 1.10i)T + (78.4 - 57.0i)T^{2} \) |
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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
−11.83044510243130953340257679686, −10.07248897338429527962613397484, −9.569371248977607518447184109280, −9.161524204344179274851957958154, −7.941460516931115327229536323718, −6.87657118687377363931483206040, −6.25783388019044423634572156950, −4.74810172251744080683851900567, −2.88053964522785571994856718953, −1.40594232072506694832309637021,
1.82033280523315399214067323847, 3.12432487235205620287049352720, 3.91316546338010747421964303157, 6.15930660478374720440594299077, 6.98410532673609873792023107265, 8.336120558983057414405625420760, 9.149089217828868412724116210213, 9.809724732210421309960762013807, 10.42004001770529354656311229365, 11.56199775381683320381044658338
