L(s) = 1 | + (1.5 + 2.59i)2-s + (−0.5 + 0.866i)4-s + 9·5-s + (−1 + 1.73i)7-s + 21·8-s + (13.5 + 23.3i)10-s + (15 + 25.9i)11-s + (32.5 − 33.7i)13-s − 6·14-s + (35.5 + 61.4i)16-s + (−55.5 + 96.1i)17-s + (23 − 39.8i)19-s + (−4.5 + 7.79i)20-s + (−45 + 77.9i)22-s + (−3 − 5.19i)23-s + ⋯ |
L(s) = 1 | + (0.530 + 0.918i)2-s + (−0.0625 + 0.108i)4-s + 0.804·5-s + (−0.0539 + 0.0935i)7-s + 0.928·8-s + (0.426 + 0.739i)10-s + (0.411 + 0.712i)11-s + (0.693 − 0.720i)13-s − 0.114·14-s + (0.554 + 0.960i)16-s + (−0.791 + 1.37i)17-s + (0.277 − 0.481i)19-s + (−0.0503 + 0.0871i)20-s + (−0.436 + 0.755i)22-s + (−0.0271 − 0.0471i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.22183 + 1.32105i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.22183 + 1.32105i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 + (-32.5 + 33.7i)T \) |
good | 2 | \( 1 + (-1.5 - 2.59i)T + (-4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 9T + 125T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-15 - 25.9i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 17 | \( 1 + (55.5 - 96.1i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-23 + 39.8i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (52.5 + 90.9i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + 100T + 2.97e4T^{2} \) |
| 37 | \( 1 + (8.5 + 14.7i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (115.5 + 200. i)T + (-3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-257 + 445. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 - 162T + 1.03e5T^{2} \) |
| 53 | \( 1 + 639T + 1.48e5T^{2} \) |
| 59 | \( 1 + (-300 + 519. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (116.5 - 201. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (463 + 801. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (465 - 805. i)T + (-1.78e5 - 3.09e5i)T^{2} \) |
| 73 | \( 1 + 253T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 810T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-249 - 431. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (679 - 1.17e3i)T + (-4.56e5 - 7.90e5i)T^{2} \) |
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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
−13.43210084596291737933072821190, −12.63580747924479550990377985281, −11.00456295744853967184249723267, −10.10679819391296597530508796590, −8.824106990181890564379696977260, −7.46009440396919559274862177859, −6.30277198921207241574782966663, −5.56085498300444515836785084161, −4.11761748298800714492538725836, −1.83443535941580837755260848413,
1.56714298362230859634875059950, 3.04725332939264108393957179846, 4.41192177725563983346359950072, 5.91482495112269628691916948868, 7.24866087342947947889409969194, 8.855199398443890316467726803264, 9.889836037190048215400670228634, 11.13650096035384196506808555589, 11.67303944268787283790913006182, 12.98181605368131781045299577396