| L(s) = 1 | + (−1.64 + 1.53i)2-s + (0.953 − 0.143i)3-s + (−2.01 + 26.8i)4-s + (28.0 − 71.4i)5-s + (−1.35 + 1.69i)6-s + (123. + 38.9i)7-s + (−82.6 − 103. i)8-s + (−231. + 71.3i)9-s + (63.1 + 160. i)10-s + (724. + 223. i)11-s + (1.94 + 25.9i)12-s + (154. + 678. i)13-s + (−263. + 125. i)14-s + (16.4 − 72.2i)15-s + (−557. − 83.9i)16-s + (1.04e3 − 709. i)17-s + ⋯ |
| L(s) = 1 | + (−0.291 + 0.270i)2-s + (0.0611 − 0.00921i)3-s + (−0.0629 + 0.839i)4-s + (0.501 − 1.27i)5-s + (−0.0153 + 0.0192i)6-s + (0.953 + 0.300i)7-s + (−0.456 − 0.572i)8-s + (−0.951 + 0.293i)9-s + (0.199 + 0.508i)10-s + (1.80 + 0.556i)11-s + (0.00389 + 0.0519i)12-s + (0.254 + 1.11i)13-s + (−0.359 + 0.170i)14-s + (0.0189 − 0.0828i)15-s + (−0.544 − 0.0820i)16-s + (0.873 − 0.595i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.56573 + 0.642667i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.56573 + 0.642667i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-123. - 38.9i)T \) |
| good | 2 | \( 1 + (1.64 - 1.53i)T + (2.39 - 31.9i)T^{2} \) |
| 3 | \( 1 + (-0.953 + 0.143i)T + (232. - 71.6i)T^{2} \) |
| 5 | \( 1 + (-28.0 + 71.4i)T + (-2.29e3 - 2.12e3i)T^{2} \) |
| 11 | \( 1 + (-724. - 223. i)T + (1.33e5 + 9.07e4i)T^{2} \) |
| 13 | \( 1 + (-154. - 678. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-1.04e3 + 709. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (-1.15e3 - 2.00e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-755. - 515. i)T + (2.35e6 + 5.99e6i)T^{2} \) |
| 29 | \( 1 + (4.66e3 + 2.24e3i)T + (1.27e7 + 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-430. + 744. i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (736. + 9.82e3i)T + (-6.85e7 + 1.03e7i)T^{2} \) |
| 41 | \( 1 + (8.12e3 + 1.01e4i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 43 | \( 1 + (-1.35e3 + 1.69e3i)T + (-3.27e7 - 1.43e8i)T^{2} \) |
| 47 | \( 1 + (-5.62e3 + 5.21e3i)T + (1.71e7 - 2.28e8i)T^{2} \) |
| 53 | \( 1 + (1.82e3 - 2.44e4i)T + (-4.13e8 - 6.23e7i)T^{2} \) |
| 59 | \( 1 + (-3.68e3 - 9.37e3i)T + (-5.24e8 + 4.86e8i)T^{2} \) |
| 61 | \( 1 + (-1.97e3 - 2.63e4i)T + (-8.35e8 + 1.25e8i)T^{2} \) |
| 67 | \( 1 + (-5.48e3 + 9.50e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (2.84e4 - 1.36e4i)T + (1.12e9 - 1.41e9i)T^{2} \) |
| 73 | \( 1 + (5.43e4 + 5.04e4i)T + (1.54e8 + 2.06e9i)T^{2} \) |
| 79 | \( 1 + (3.19e4 + 5.54e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-8.86e3 + 3.88e4i)T + (-3.54e9 - 1.70e9i)T^{2} \) |
| 89 | \( 1 + (6.56e4 - 2.02e4i)T + (4.61e9 - 3.14e9i)T^{2} \) |
| 97 | \( 1 - 1.72e4T + 8.58e9T^{2} \) |
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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
−14.56223479746104544996010660906, −13.73159967222439492532144003611, −12.06682241336824671219880278403, −11.79048782487313535201120817330, −9.271062132687432382139131043472, −8.842696826932082895950165544571, −7.51639298562246718298634943774, −5.64947137372004223771071437606, −4.08733154727364978964881298386, −1.57165342267351320590558546473,
1.19068804127498725199807105926, 3.13302693027990897992945854655, 5.54334821403819855488748443491, 6.66618302191498168694427081691, 8.534293438153149955110870572384, 9.804712103446252745384050953716, 11.02110395551611283800697838524, 11.50152451890703799358230374579, 13.80214530066725528509368651638, 14.55986423420422194231358145326
