| L(s) = 1 | + (−2.52e3 − 4.37e3i)5-s + (1.39e4 − 4.22e4i)7-s + (−7.03e4 − 4.06e4i)11-s − 1.76e6i·13-s + (9.41e5 − 1.63e6i)17-s + (−5.33e6 + 3.07e6i)19-s + (4.10e7 − 2.37e7i)23-s + (1.16e7 − 2.01e7i)25-s − 1.21e8i·29-s + (−1.21e8 − 7.03e7i)31-s + (−2.20e8 + 4.58e7i)35-s + (6.03e7 + 1.04e8i)37-s + 8.24e8·41-s + 1.15e9·43-s + (−1.30e9 − 2.26e9i)47-s + ⋯ |
| L(s) = 1 | + (−0.361 − 0.626i)5-s + (0.312 − 0.949i)7-s + (−0.131 − 0.0760i)11-s − 1.32i·13-s + (0.160 − 0.278i)17-s + (−0.493 + 0.285i)19-s + (1.32 − 0.767i)23-s + (0.238 − 0.412i)25-s − 1.10i·29-s + (−0.764 − 0.441i)31-s + (−0.708 + 0.147i)35-s + (0.143 + 0.247i)37-s + 1.11·41-s + 1.19·43-s + (−0.832 − 1.44i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.978120968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.978120968\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + (-1.39e4 + 4.22e4i)T \) |
| good | 5 | \( 1 + (2.52e3 + 4.37e3i)T + (-2.44e7 + 4.22e7i)T^{2} \) |
| 11 | \( 1 + (7.03e4 + 4.06e4i)T + (1.42e11 + 2.47e11i)T^{2} \) |
| 13 | \( 1 + 1.76e6iT - 1.79e12T^{2} \) |
| 17 | \( 1 + (-9.41e5 + 1.63e6i)T + (-1.71e13 - 2.96e13i)T^{2} \) |
| 19 | \( 1 + (5.33e6 - 3.07e6i)T + (5.82e13 - 1.00e14i)T^{2} \) |
| 23 | \( 1 + (-4.10e7 + 2.37e7i)T + (4.76e14 - 8.25e14i)T^{2} \) |
| 29 | \( 1 + 1.21e8iT - 1.22e16T^{2} \) |
| 31 | \( 1 + (1.21e8 + 7.03e7i)T + (1.27e16 + 2.20e16i)T^{2} \) |
| 37 | \( 1 + (-6.03e7 - 1.04e8i)T + (-8.89e16 + 1.54e17i)T^{2} \) |
| 41 | \( 1 - 8.24e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.15e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + (1.30e9 + 2.26e9i)T + (-1.23e18 + 2.14e18i)T^{2} \) |
| 53 | \( 1 + (-1.12e9 - 6.49e8i)T + (4.63e18 + 8.02e18i)T^{2} \) |
| 59 | \( 1 + (-3.94e9 + 6.82e9i)T + (-1.50e19 - 2.61e19i)T^{2} \) |
| 61 | \( 1 + (-6.27e9 + 3.62e9i)T + (2.17e19 - 3.76e19i)T^{2} \) |
| 67 | \( 1 + (6.20e8 - 1.07e9i)T + (-6.10e19 - 1.05e20i)T^{2} \) |
| 71 | \( 1 - 1.18e10iT - 2.31e20T^{2} \) |
| 73 | \( 1 + (-2.19e10 - 1.26e10i)T + (1.56e20 + 2.71e20i)T^{2} \) |
| 79 | \( 1 + (-5.60e9 - 9.70e9i)T + (-3.73e20 + 6.47e20i)T^{2} \) |
| 83 | \( 1 + 3.86e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + (-1.46e10 - 2.53e10i)T + (-1.38e21 + 2.40e21i)T^{2} \) |
| 97 | \( 1 + 3.47e10iT - 7.15e21T^{2} \) |
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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
−9.760707331126702021826803059301, −8.484710637933971448781871909329, −7.87586652501357331163652253403, −6.86688738027272204943874176584, −5.54752588808376183357667639720, −4.61054512846632607153653405520, −3.68613264001659463761312961426, −2.42241089481664767384144559475, −0.860322586436917344358434281281, −0.46789148639822591047432390211,
1.28952910457718938567071148867, 2.35580425961931404199889330276, 3.36576828318073647688305996233, 4.57904430199814457195269923911, 5.62226115506112778572026472334, 6.76078282648383795306079027271, 7.52870093849139277698686653176, 8.833968933504933196847215588133, 9.336493712614424151409747365500, 10.86028405044736923056053868611
