| L(s) = 1 | + (447. − 775. i)2-s + (−7.85e3 + 4.53e3i)3-s + (−2.69e5 − 4.66e5i)4-s + (−2.15e6 − 1.24e6i)5-s + 8.12e6i·6-s + (3.85e7 + 1.19e7i)7-s − 2.47e8·8-s + (−1.52e8 + 2.64e8i)9-s + (−1.92e9 + 1.11e9i)10-s + (−1.02e9 − 1.77e9i)11-s + (4.23e9 + 2.44e9i)12-s + 1.33e10i·13-s + (2.65e10 − 2.45e10i)14-s + 2.25e10·15-s + (−4.01e10 + 6.95e10i)16-s + (−6.78e10 + 3.91e10i)17-s + ⋯ |
| L(s) = 1 | + (0.873 − 1.51i)2-s + (−0.399 + 0.230i)3-s + (−1.02 − 1.77i)4-s + (−1.10 − 0.636i)5-s + 0.805i·6-s + (0.955 + 0.296i)7-s − 1.84·8-s + (−0.393 + 0.681i)9-s + (−1.92 + 1.11i)10-s + (−0.434 − 0.752i)11-s + (0.820 + 0.473i)12-s + 1.26i·13-s + (1.28 − 1.18i)14-s + 0.587·15-s + (−0.584 + 1.01i)16-s + (−0.572 + 0.330i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0957 - 0.995i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.0957 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(0.379609 + 0.417860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.379609 + 0.417860i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-3.85e7 - 1.19e7i)T \) |
| good | 2 | \( 1 + (-447. + 775. i)T + (-1.31e5 - 2.27e5i)T^{2} \) |
| 3 | \( 1 + (7.85e3 - 4.53e3i)T + (1.93e8 - 3.35e8i)T^{2} \) |
| 5 | \( 1 + (2.15e6 + 1.24e6i)T + (1.90e12 + 3.30e12i)T^{2} \) |
| 11 | \( 1 + (1.02e9 + 1.77e9i)T + (-2.77e18 + 4.81e18i)T^{2} \) |
| 13 | \( 1 - 1.33e10iT - 1.12e20T^{2} \) |
| 17 | \( 1 + (6.78e10 - 3.91e10i)T + (7.03e21 - 1.21e22i)T^{2} \) |
| 19 | \( 1 + (2.45e11 + 1.41e11i)T + (5.20e22 + 9.01e22i)T^{2} \) |
| 23 | \( 1 + (-9.69e11 + 1.67e12i)T + (-1.62e24 - 2.80e24i)T^{2} \) |
| 29 | \( 1 + 2.51e13T + 2.10e26T^{2} \) |
| 31 | \( 1 + (1.72e13 - 9.93e12i)T + (3.49e26 - 6.05e26i)T^{2} \) |
| 37 | \( 1 + (-6.86e13 + 1.18e14i)T + (-8.44e27 - 1.46e28i)T^{2} \) |
| 41 | \( 1 - 1.61e14iT - 1.07e29T^{2} \) |
| 43 | \( 1 + 7.27e14T + 2.52e29T^{2} \) |
| 47 | \( 1 + (5.15e14 + 2.97e14i)T + (6.26e29 + 1.08e30i)T^{2} \) |
| 53 | \( 1 + (1.00e15 + 1.73e15i)T + (-5.44e30 + 9.42e30i)T^{2} \) |
| 59 | \( 1 + (-9.23e15 + 5.33e15i)T + (3.75e31 - 6.49e31i)T^{2} \) |
| 61 | \( 1 + (-1.91e16 - 1.10e16i)T + (6.83e31 + 1.18e32i)T^{2} \) |
| 67 | \( 1 + (7.08e15 + 1.22e16i)T + (-3.70e32 + 6.41e32i)T^{2} \) |
| 71 | \( 1 - 5.99e16T + 2.10e33T^{2} \) |
| 73 | \( 1 + (1.65e16 - 9.54e15i)T + (1.73e33 - 3.00e33i)T^{2} \) |
| 79 | \( 1 + (1.39e16 - 2.41e16i)T + (-7.18e33 - 1.24e34i)T^{2} \) |
| 83 | \( 1 + 2.09e17iT - 3.49e34T^{2} \) |
| 89 | \( 1 + (3.04e17 + 1.75e17i)T + (6.13e34 + 1.06e35i)T^{2} \) |
| 97 | \( 1 + 1.12e15iT - 5.77e35T^{2} \) |
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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
−16.50615236623587244535893372945, −14.64296529906475047819814979057, −13.03576873005711059686166904852, −11.50193089863398189686613551279, −11.05334700809730519960237528538, −8.579163568811918431820424739029, −5.12216081778155795282358399165, −4.12120640885687507153957523990, −2.05454267007584233446790199767, −0.17823497149420602430401952195,
3.72506072550929149047287492280, 5.32367782807799343627388631498, 7.06306672236915329789575395830, 8.022611522713933706058408134022, 11.33830156402688825281747785748, 12.94237688047547062542421696049, 14.88300765460684930225681555819, 15.24653924198520649120417671707, 17.11191870452160999165256125922, 18.14380239202856196558735881035
