| L(s) = 1 | + (−0.819 + 0.573i)2-s + (0.342 − 0.939i)4-s + (0.463 + 2.18i)5-s + (−1.78 + 3.83i)7-s + (0.258 + 0.965i)8-s + (−1.63 − 1.52i)10-s + (−3.34 + 3.98i)11-s + (−0.215 + 0.307i)13-s + (−0.734 − 4.16i)14-s + (−0.766 − 0.642i)16-s + (1.58 − 5.90i)17-s + (−2.93 − 1.69i)19-s + (2.21 + 0.313i)20-s + (0.453 − 5.18i)22-s + (1.01 − 0.472i)23-s + ⋯ |
| L(s) = 1 | + (−0.579 + 0.405i)2-s + (0.171 − 0.469i)4-s + (0.207 + 0.978i)5-s + (−0.675 + 1.44i)7-s + (0.0915 + 0.341i)8-s + (−0.516 − 0.482i)10-s + (−1.00 + 1.20i)11-s + (−0.0597 + 0.0853i)13-s + (−0.196 − 1.11i)14-s + (−0.191 − 0.160i)16-s + (0.383 − 1.43i)17-s + (−0.672 − 0.388i)19-s + (0.495 + 0.0699i)20-s + (0.0966 − 1.10i)22-s + (0.211 − 0.0984i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0991386 - 0.499609i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0991386 - 0.499609i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.819 - 0.573i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.463 - 2.18i)T \) |
| good | 7 | \( 1 + (1.78 - 3.83i)T + (-4.49 - 5.36i)T^{2} \) |
| 11 | \( 1 + (3.34 - 3.98i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (0.215 - 0.307i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.58 + 5.90i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (2.93 + 1.69i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.01 + 0.472i)T + (14.7 - 17.6i)T^{2} \) |
| 29 | \( 1 + (-1.75 + 9.97i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (3.35 + 1.22i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-5.63 - 1.51i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (4.77 - 0.841i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.570 - 6.52i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-6.65 - 3.10i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (6.39 - 6.39i)T - 53iT^{2} \) |
| 59 | \( 1 + (2.30 - 1.93i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (7.18 - 2.61i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.199 - 0.139i)T + (22.9 + 62.9i)T^{2} \) |
| 71 | \( 1 + (0.0530 - 0.0306i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.38 + 1.71i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (5.15 + 0.909i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.73 - 3.91i)T + (-28.3 + 77.9i)T^{2} \) |
| 89 | \( 1 + (4.63 - 8.03i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (15.6 - 1.36i)T + (95.5 - 16.8i)T^{2} \) |
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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
−10.50062091444022883625226633288, −9.569056100548080844202579974806, −9.399148203610060075264515271513, −8.038683850167669878460039027352, −7.33656739420198685327395682696, −6.43098271836976724208407696201, −5.72280997548444441609298070375, −4.67517565683422895296795479704, −2.76700180384748269940010781461, −2.36433220036074567903724355579,
0.29703709832105443623943547079, 1.53017101833610254742246938993, 3.25334745533510946046213905756, 4.02841969550506147613423260540, 5.31912736652242879842041585557, 6.32143976178263239151031708911, 7.42403631237463012606091293450, 8.268630408576225880771035712514, 8.841165264948051663925892203258, 9.961717381420606854400458952789
