| L(s) = 1 | + (−39.4 + 25.0i)3-s + (−137. + 238. i)5-s + (171.5 + 297. i)7-s + (928. − 1.98e3i)9-s + (1.72e3 + 2.99e3i)11-s + (1.83e3 − 3.17e3i)13-s + (−548. − 1.28e4i)15-s − 4.87e3·17-s − 4.01e4·19-s + (−1.42e4 − 7.42e3i)21-s + (−5.24e4 + 9.08e4i)23-s + (1.05e3 + 1.81e3i)25-s + (1.30e4 + 1.01e5i)27-s + (−1.59e4 − 2.76e4i)29-s + (−222. + 384. i)31-s + ⋯ |
| L(s) = 1 | + (−0.843 + 0.536i)3-s + (−0.493 + 0.854i)5-s + (0.188 + 0.327i)7-s + (0.424 − 0.905i)9-s + (0.391 + 0.678i)11-s + (0.231 − 0.400i)13-s + (−0.0420 − 0.985i)15-s − 0.240·17-s − 1.34·19-s + (−0.335 − 0.174i)21-s + (−0.898 + 1.55i)23-s + (0.0134 + 0.0232i)25-s + (0.127 + 0.991i)27-s + (−0.121 − 0.210i)29-s + (−0.00133 + 0.00231i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.07966933400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07966933400\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (39.4 - 25.0i)T \) |
| 7 | \( 1 + (-171.5 - 297. i)T \) |
| good | 5 | \( 1 + (137. - 238. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (-1.72e3 - 2.99e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-1.83e3 + 3.17e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + 4.87e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.01e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + (5.24e4 - 9.08e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.59e4 + 2.76e4i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 + (222. - 384. i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.55e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.25e5 - 2.17e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (6.08e4 + 1.05e5i)T + (-1.35e11 + 2.35e11i)T^{2} \) |
| 47 | \( 1 + (1.61e5 + 2.78e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 - 1.10e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + (4.01e5 - 6.94e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.68e5 + 2.92e5i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.45e5 - 4.25e5i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 + 2.10e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.64e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + (2.08e5 + 3.60e5i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 + (3.34e6 + 5.79e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 - 6.64e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (5.90e6 + 1.02e7i)T + (-4.03e13 + 6.99e13i)T^{2} \) |
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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
−10.65702140395782018591922368924, −9.906055995298427208656207600057, −8.756756774853994531098556589116, −7.43972104191848646122202175651, −6.54331345192000885567566815772, −5.55056300704063466889326481925, −4.32490064651485380447348885323, −3.40092678268323969975506648788, −1.77172188107172607892893826015, −0.02742580741355789381062988460,
0.821811367684440986932150709961, 2.03034825360359359578204126497, 4.02287317812146503441470280682, 4.80150049107905909892487241282, 6.08196242511453800933177920286, 6.87802079321306946935173072094, 8.195583356507811150426030250695, 8.748825666049932859383291382380, 10.34611496972009407630972375352, 11.07514504255465566627734938443
