| L(s) = 1 | + (−10.7 + 3.49i)2-s + (−42.1 − 30.6i)3-s + (103. − 75.2i)4-s + (−302. + 929. i)5-s + (560. + 182. i)6-s + (−997. − 1.37e3i)7-s + (−851. + 1.17e3i)8-s + (−1.18e3 − 3.66e3i)9-s − 1.10e4i·10-s + (1.44e4 + 2.17e3i)11-s − 6.66e3·12-s + (5.03e4 − 1.63e4i)13-s + (1.55e4 + 1.12e4i)14-s + (4.11e4 − 2.99e4i)15-s + (5.06e3 − 1.55e4i)16-s + (−1.33e5 − 4.32e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.520 − 0.377i)3-s + (0.404 − 0.293i)4-s + (−0.483 + 1.48i)5-s + (0.432 + 0.140i)6-s + (−0.415 − 0.571i)7-s + (−0.207 + 0.286i)8-s + (−0.181 − 0.557i)9-s − 1.10i·10-s + (0.988 + 0.148i)11-s − 0.321·12-s + (1.76 − 0.573i)13-s + (0.404 + 0.293i)14-s + (0.813 − 0.591i)15-s + (0.0772 − 0.237i)16-s + (−1.59 − 0.517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.771073 - 0.305955i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.771073 - 0.305955i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (10.7 - 3.49i)T \) |
| 11 | \( 1 + (-1.44e4 - 2.17e3i)T \) |
| good | 3 | \( 1 + (42.1 + 30.6i)T + (2.02e3 + 6.23e3i)T^{2} \) |
| 5 | \( 1 + (302. - 929. i)T + (-3.16e5 - 2.29e5i)T^{2} \) |
| 7 | \( 1 + (997. + 1.37e3i)T + (-1.78e6 + 5.48e6i)T^{2} \) |
| 13 | \( 1 + (-5.03e4 + 1.63e4i)T + (6.59e8 - 4.79e8i)T^{2} \) |
| 17 | \( 1 + (1.33e5 + 4.32e4i)T + (5.64e9 + 4.10e9i)T^{2} \) |
| 19 | \( 1 + (-1.20e5 + 1.65e5i)T + (-5.24e9 - 1.61e10i)T^{2} \) |
| 23 | \( 1 - 2.54e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (-2.77e5 - 3.82e5i)T + (-1.54e11 + 4.75e11i)T^{2} \) |
| 31 | \( 1 + (2.88e4 + 8.87e4i)T + (-6.90e11 + 5.01e11i)T^{2} \) |
| 37 | \( 1 + (-4.15e5 + 3.01e5i)T + (1.08e12 - 3.34e12i)T^{2} \) |
| 41 | \( 1 + (-8.14e5 + 1.12e6i)T + (-2.46e12 - 7.59e12i)T^{2} \) |
| 43 | \( 1 - 4.18e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 + (-1.08e6 - 7.91e5i)T + (7.35e12 + 2.26e13i)T^{2} \) |
| 53 | \( 1 + (4.16e6 + 1.28e7i)T + (-5.03e13 + 3.65e13i)T^{2} \) |
| 59 | \( 1 + (-4.73e6 + 3.43e6i)T + (4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-5.32e6 - 1.73e6i)T + (1.55e14 + 1.12e14i)T^{2} \) |
| 67 | \( 1 + 2.24e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + (-9.36e6 + 2.88e7i)T + (-5.22e14 - 3.79e14i)T^{2} \) |
| 73 | \( 1 + (1.11e7 + 1.54e7i)T + (-2.49e14 + 7.66e14i)T^{2} \) |
| 79 | \( 1 + (-4.26e6 + 1.38e6i)T + (1.22e15 - 8.91e14i)T^{2} \) |
| 83 | \( 1 + (-4.95e7 - 1.60e7i)T + (1.82e15 + 1.32e15i)T^{2} \) |
| 89 | \( 1 - 1.62e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + (1.43e7 + 4.42e7i)T + (-6.34e15 + 4.60e15i)T^{2} \) |
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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
−15.99559808606620117963540238388, −14.99213299807312865858935861089, −13.49514325857994134203301150268, −11.43786299920898418704333911001, −10.89016771523017597501496783256, −9.070248908649223907691810900607, −7.01108171154720652675178453904, −6.49288556883823354548543727114, −3.34598206364849066461414780388, −0.65354484373531078797042419230,
1.24588040800831927520251274376, 4.17343011600355427749727630800, 6.03453499313062034897039152306, 8.417531649128801775305790171920, 9.189421554471878340946663058893, 11.07258766946302294059973944417, 12.03205417358298497862008200253, 13.40426025956194538636978828191, 15.76460961793248390339142713827, 16.32691327881300013930768364805
