| L(s) = 1 | + 1.93i·2-s + (−5.02 + 7.46i)3-s + 12.2·4-s + 4.62·5-s + (−14.4 − 9.71i)6-s + 58.0i·7-s + 54.6i·8-s + (−30.4 − 75.0i)9-s + 8.93i·10-s − 163.·11-s + (−61.7 + 91.5i)12-s − 81.9·13-s − 112.·14-s + (−23.2 + 34.5i)15-s + 90.8·16-s + (−44.7 + 285. i)17-s + ⋯ |
| L(s) = 1 | + 0.482i·2-s + (−0.558 + 0.829i)3-s + 0.766·4-s + 0.185·5-s + (−0.400 − 0.269i)6-s + 1.18i·7-s + 0.853i·8-s + (−0.375 − 0.926i)9-s + 0.0893i·10-s − 1.35·11-s + (−0.428 + 0.635i)12-s − 0.484·13-s − 0.571·14-s + (−0.103 + 0.153i)15-s + 0.354·16-s + (−0.154 + 0.987i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.499808 + 1.27247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.499808 + 1.27247i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.02 - 7.46i)T \) |
| 17 | \( 1 + (44.7 - 285. i)T \) |
| good | 2 | \( 1 - 1.93iT - 16T^{2} \) |
| 5 | \( 1 - 4.62T + 625T^{2} \) |
| 7 | \( 1 - 58.0iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 163.T + 1.46e4T^{2} \) |
| 13 | \( 1 + 81.9T + 2.85e4T^{2} \) |
| 19 | \( 1 - 524.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 767.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 462.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 637. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.17e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 - 1.29e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.44e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 3.32e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 257. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.90e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 3.60e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 1.63e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.21e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 7.48e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 2.59e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 4.31e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.09e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 4.25e3iT - 8.85e7T^{2} \) |
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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
−15.56304062251758991485231141845, −14.55891321056566950244586522955, −12.67048020863077947191220197914, −11.57714319514804609214193493105, −10.57845642988958164908705176355, −9.257384746263001408204375015697, −7.72335579437950986092518873834, −5.99452762165832188935295619099, −5.21576139148960239604632081159, −2.72434524772730665521247204486,
0.884663034484547804391313462900, 2.75352541247958758674217804420, 5.24084859977765054488359051553, 6.99016991779222361878219784313, 7.60975296355792106812219439491, 10.01117252606116301775354552055, 10.92989027636961145212684805454, 11.89397413447151594805949320885, 13.08935598590847846731398439920, 13.87458110328952453471115056992
