L(s) = 1 | + (−2.08 − 2.08i)2-s + (3.67 − 3.67i)3-s − 7.34i·4-s + (−8.43 − 23.5i)5-s − 15.2·6-s + (65.1 + 65.1i)7-s + (−48.5 + 48.5i)8-s − 27i·9-s + (−31.4 + 66.5i)10-s + 56.3·11-s + (−26.9 − 26.9i)12-s + (0.983 − 0.983i)13-s − 270. i·14-s + (−117. − 55.4i)15-s + 84.5·16-s + (159. + 159. i)17-s + ⋯ |
L(s) = 1 | + (−0.520 − 0.520i)2-s + (0.408 − 0.408i)3-s − 0.458i·4-s + (−0.337 − 0.941i)5-s − 0.424·6-s + (1.32 + 1.32i)7-s + (−0.758 + 0.758i)8-s − 0.333i·9-s + (−0.314 + 0.665i)10-s + 0.466·11-s + (−0.187 − 0.187i)12-s + (0.00582 − 0.00582i)13-s − 1.38i·14-s + (−0.522 − 0.246i)15-s + 0.330·16-s + (0.550 + 0.550i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.112 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.773033 - 0.690767i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.773033 - 0.690767i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.67 + 3.67i)T \) |
| 5 | \( 1 + (8.43 + 23.5i)T \) |
good | 2 | \( 1 + (2.08 + 2.08i)T + 16iT^{2} \) |
| 7 | \( 1 + (-65.1 - 65.1i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 - 56.3T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-0.983 + 0.983i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-159. - 159. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + 265. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (185. - 185. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 - 544. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 710.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (639. + 639. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.32e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (22.3 - 22.3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-456. - 456. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (424. - 424. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + 3.46e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.93e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (3.80e3 + 3.80e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 7.95e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (1.94e3 - 1.94e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 4.08e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (9.10e3 - 9.10e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 7.01e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-2.57e3 - 2.57e3i)T + 8.85e7iT^{2} \) |
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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
−18.49228985114559023003074496801, −17.44810255768590886003972278022, −15.41698655654696246711381111598, −14.37330894565591225522574638448, −12.36013610192514077518251305569, −11.35540306174801438388166646469, −9.185104374136714160875130453939, −8.295989096159804705762791380459, −5.36473749132794093293437779109, −1.68035869254979326225587030828,
3.87283545399608238211153861663, 7.18272803020851870699422545652, 8.176273455016128514592519486939, 10.17792517255598073271382228798, 11.62546862493771462214873354531, 13.94344395282377809556926847972, 14.89964633872149849278697181704, 16.43005607477516101637792741124, 17.48921789092751526822064674624, 18.64348454914237479934499292337