| L(s) = 1 | + (−2.73 − 0.708i)2-s + (−4.61 − 2.39i)3-s + (6.99 + 3.88i)4-s + (0.577 − 1.58i)5-s + (10.9 + 9.82i)6-s + (−27.0 − 4.76i)7-s + (−16.3 − 15.5i)8-s + (15.5 + 22.0i)9-s + (−2.70 + 3.93i)10-s + (19.7 − 7.17i)11-s + (−22.9 − 34.6i)12-s + (−48.6 + 40.8i)13-s + (70.5 + 32.1i)14-s + (−6.45 + 5.93i)15-s + (33.8 + 54.3i)16-s + (103. − 59.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.968 − 0.250i)2-s + (−0.887 − 0.460i)3-s + (0.874 + 0.485i)4-s + (0.0516 − 0.141i)5-s + (0.743 + 0.668i)6-s + (−1.45 − 0.257i)7-s + (−0.724 − 0.688i)8-s + (0.575 + 0.817i)9-s + (−0.0855 + 0.124i)10-s + (0.540 − 0.196i)11-s + (−0.552 − 0.833i)12-s + (−1.03 + 0.870i)13-s + (1.34 + 0.614i)14-s + (−0.111 + 0.102i)15-s + (0.528 + 0.848i)16-s + (1.48 − 0.855i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.485704 + 0.154877i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.485704 + 0.154877i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.73 + 0.708i)T \) |
| 3 | \( 1 + (4.61 + 2.39i)T \) |
| good | 5 | \( 1 + (-0.577 + 1.58i)T + (-95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (27.0 + 4.76i)T + (322. + 117. i)T^{2} \) |
| 11 | \( 1 + (-19.7 + 7.17i)T + (1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (48.6 - 40.8i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (-103. + 59.9i)T + (2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-110. - 63.5i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-21.9 - 124. i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (3.97 - 4.73i)T + (-4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (182. - 32.1i)T + (2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-34.7 - 60.1i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-77.0 - 91.8i)T + (-1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (16.6 + 45.7i)T + (-6.09e4 + 5.11e4i)T^{2} \) |
| 47 | \( 1 + (87.1 - 493. i)T + (-9.75e4 - 3.55e4i)T^{2} \) |
| 53 | \( 1 + 387. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (226. + 82.3i)T + (1.57e5 + 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-5.95 + 33.7i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-320. - 381. i)T + (-5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-128. - 221. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (401. - 694. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (41.6 - 49.6i)T + (-8.56e4 - 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-814. - 683. i)T + (9.92e4 + 5.63e5i)T^{2} \) |
| 89 | \( 1 + (741. + 427. i)T + (3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (805. - 293. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−12.86302321529191324373713433357, −12.10309928271104680939312395001, −11.32049394866880247861709097223, −9.783224510611505609080735130092, −9.550878078366653954626262802813, −7.52011827851190746267411297486, −6.90034255908876086065974018713, −5.58993184686187916034076547471, −3.28832745061949563095412293541, −1.15376586645389858566343256145,
0.51032008849704596882790304676, 3.14138324051033552477372657523, 5.38715532611491400082980343885, 6.40158876987361145700654782530, 7.40283688974773263803396534710, 9.161543707555585558915249543690, 9.935410064060850177415246985448, 10.59767890969707239721033856950, 12.05876387135382061052989746458, 12.61143083942658474510567540914
