L(s) = 1 | + (1.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (3 − 1.73i)5-s − 1.73i·6-s + (−0.5 + 2.59i)7-s + 1.73i·8-s + (−0.499 − 0.866i)9-s + (3 − 5.19i)10-s + (−4.5 − 2.59i)11-s + (−0.500 − 0.866i)12-s + (−1 + 3.46i)13-s + (1.5 + 4.33i)14-s − 3.46i·15-s + (2.49 + 4.33i)16-s + (1.5 − 2.59i)17-s + ⋯ |
L(s) = 1 | + (1.06 − 0.612i)2-s + (0.288 − 0.499i)3-s + (0.250 − 0.433i)4-s + (1.34 − 0.774i)5-s − 0.707i·6-s + (−0.188 + 0.981i)7-s + 0.612i·8-s + (−0.166 − 0.288i)9-s + (0.948 − 1.64i)10-s + (−1.35 − 0.783i)11-s + (−0.144 − 0.249i)12-s + (−0.277 + 0.960i)13-s + (0.400 + 1.15i)14-s − 0.894i·15-s + (0.624 + 1.08i)16-s + (0.363 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.490 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.13650 - 1.24914i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.13650 - 1.24914i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
| 13 | \( 1 + (1 - 3.46i)T \) |
good | 2 | \( 1 + (-1.5 + 0.866i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.5 - 2.59i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 + 1.73i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-1.5 + 0.866i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.5 - 4.33i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 0.866i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.1iT - 71T^{2} \) |
| 73 | \( 1 + (-9 - 5.19i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (3 - 1.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 17.3iT - 97T^{2} \) |
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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
−12.24890137542472350220866597875, −11.07190447606080138406345064927, −9.877489647000114415312745183109, −8.848045810380763734719560423030, −8.117348889148761797760378546942, −6.19057013816644608343017777255, −5.59948891776566973267832108567, −4.57101742365766391650417719538, −2.74489311513659311615471856407, −2.07685404008528896978942627280,
2.50665517112265774253943274515, 3.79705292345275806543833505104, 5.07036367408798051558518761101, 5.83359462454362716338919437435, 6.92344577144066472899688655164, 7.80493474134101319127737302664, 9.573669773150660070830699850600, 10.31803819795874944537358884081, 10.62086334778531193934628477378, 12.70187410241916295868444410456