L(s) = 1 | + (−9.74 − 16.8i)5-s + (−18.4 + 1.67i)7-s + (−15.1 − 8.72i)11-s − 16.5i·13-s + (68.6 − 118. i)17-s + (−20.6 + 11.9i)19-s + (−108. + 62.8i)23-s + (−127. + 220. i)25-s + 105. i·29-s + (−95.0 − 54.8i)31-s + (207. + 294. i)35-s + (58.5 + 101. i)37-s + 348.·41-s − 141.·43-s + (172. + 299. i)47-s + ⋯ |
L(s) = 1 | + (−0.871 − 1.50i)5-s + (−0.995 + 0.0902i)7-s + (−0.414 − 0.239i)11-s − 0.352i·13-s + (0.979 − 1.69i)17-s + (−0.249 + 0.144i)19-s + (−0.987 + 0.570i)23-s + (−1.01 + 1.76i)25-s + 0.673i·29-s + (−0.550 − 0.318i)31-s + (1.00 + 1.42i)35-s + (0.260 + 0.450i)37-s + 1.32·41-s − 0.503·43-s + (0.536 + 0.929i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.124 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.1052535662\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1052535662\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (18.4 - 1.67i)T \) |
good | 5 | \( 1 + (9.74 + 16.8i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (15.1 + 8.72i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 16.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-68.6 + 118. i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (20.6 - 11.9i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (108. - 62.8i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 105. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (95.0 + 54.8i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-58.5 - 101. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 141.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-172. - 299. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-149. - 86.1i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (297. - 515. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (561. - 324. i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (64.9 - 112. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 908. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (44.8 + 25.9i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-474. - 822. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (210. + 364. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 553. iT - 9.12e5T^{2} \) |
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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
−10.75894610876715449195774195720, −9.564523389002662931452422763502, −9.116239403727587123262004274711, −7.972725560621998801225338585701, −7.41478198686199793206210039278, −5.89903356430026822608571626403, −5.10131213435572787511298735228, −4.01738752081339880088602892242, −2.93946659423737329553369789290, −0.984738921736881758344738113520,
0.03974704171202348532951546776, 2.29681401642471427892755360944, 3.44634814756469440176387929394, 4.08479681774101290510118190951, 5.92670750767398190646781665775, 6.58563939091580267630267298754, 7.51411526812717490305875858953, 8.230535582229792706933217654162, 9.625214005389821017963367459562, 10.44586160214402138389193331497