L(s) = 1 | + (3.88 − 0.947i)2-s − 5.19i·3-s + (14.2 − 7.36i)4-s + 11.1·5-s + (−4.92 − 20.1i)6-s − 12.7i·7-s + (48.2 − 42.0i)8-s − 27·9-s + (43.4 − 10.5i)10-s − 45.0i·11-s + (−38.2 − 73.8i)12-s + 1.08·13-s + (−12.1 − 49.6i)14-s − 58.0i·15-s + (147. − 209. i)16-s + 40.6·17-s + ⋯ |
L(s) = 1 | + (0.971 − 0.236i)2-s − 0.577i·3-s + (0.887 − 0.460i)4-s + 0.447·5-s + (−0.136 − 0.560i)6-s − 0.260i·7-s + (0.753 − 0.657i)8-s − 0.333·9-s + (0.434 − 0.105i)10-s − 0.372i·11-s + (−0.265 − 0.512i)12-s + 0.00642·13-s + (−0.0617 − 0.253i)14-s − 0.258i·15-s + (0.576 − 0.817i)16-s + 0.140·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.49776 - 1.51842i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.49776 - 1.51842i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.88 + 0.947i)T \) |
| 3 | \( 1 + 5.19iT \) |
| 5 | \( 1 - 11.1T \) |
good | 7 | \( 1 + 12.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 45.0iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 1.08T + 2.85e4T^{2} \) |
| 17 | \( 1 - 40.6T + 8.35e4T^{2} \) |
| 19 | \( 1 - 290. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 949. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 402.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 762. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.32e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 3.20e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.38e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 730. iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.35e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 5.21e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.69e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 3.35e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 8.48e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 174.T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.01e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 8.83e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.93e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.42e4T + 8.85e7T^{2} \) |
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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
−13.87936487787347517065956405160, −13.19210517393561134321038991737, −12.06488445530554550947789959141, −11.01862295804187140873592573151, −9.710935658991049378763283024871, −7.80350187639481870410347687230, −6.47303764519994594022603624640, −5.30870883043610224937328708074, −3.40941492923467590884596407696, −1.58161136103168396973474488632,
2.54591224914774597710772893055, 4.31078391859186359851955494739, 5.55724386042459026711377538161, 6.88484091130025768846735956681, 8.557832123035008400711666889020, 10.07812608993209672425029356471, 11.25642552242024170940651068518, 12.44982970861798494275631961535, 13.51020494543651509566713643194, 14.62570320775701369162768083690