| L(s) = 1 | + (35.7 − 62.0i)2-s + (−397. + 229. i)3-s + (−515. − 892. i)4-s + (−2.36e4 − 1.36e4i)5-s + 3.28e4i·6-s + (−9.20e4 + 7.32e4i)7-s + 2.19e5·8-s + (−1.60e5 + 2.77e5i)9-s + (−1.69e6 + 9.77e5i)10-s + (−8.46e5 − 1.46e6i)11-s + (4.09e5 + 2.36e5i)12-s − 3.98e6i·13-s + (1.25e6 + 8.33e6i)14-s + 1.25e7·15-s + (9.96e6 − 1.72e7i)16-s + (6.97e6 − 4.02e6i)17-s + ⋯ |
| L(s) = 1 | + (0.559 − 0.968i)2-s + (−0.545 + 0.314i)3-s + (−0.125 − 0.217i)4-s + (−1.51 − 0.873i)5-s + 0.704i·6-s + (−0.782 + 0.623i)7-s + 0.837·8-s + (−0.301 + 0.522i)9-s + (−1.69 + 0.977i)10-s + (−0.477 − 0.827i)11-s + (0.137 + 0.0791i)12-s − 0.824i·13-s + (0.166 + 1.10i)14-s + 1.09·15-s + (0.594 − 1.02i)16-s + (0.288 − 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(0.0833458 + 0.325501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0833458 + 0.325501i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (9.20e4 - 7.32e4i)T \) |
| good | 2 | \( 1 + (-35.7 + 62.0i)T + (-2.04e3 - 3.54e3i)T^{2} \) |
| 3 | \( 1 + (397. - 229. i)T + (2.65e5 - 4.60e5i)T^{2} \) |
| 5 | \( 1 + (2.36e4 + 1.36e4i)T + (1.22e8 + 2.11e8i)T^{2} \) |
| 11 | \( 1 + (8.46e5 + 1.46e6i)T + (-1.56e12 + 2.71e12i)T^{2} \) |
| 13 | \( 1 + 3.98e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (-6.97e6 + 4.02e6i)T + (2.91e14 - 5.04e14i)T^{2} \) |
| 19 | \( 1 + (4.22e7 + 2.43e7i)T + (1.10e15 + 1.91e15i)T^{2} \) |
| 23 | \( 1 + (2.38e7 - 4.12e7i)T + (-1.09e16 - 1.89e16i)T^{2} \) |
| 29 | \( 1 - 4.66e8T + 3.53e17T^{2} \) |
| 31 | \( 1 + (1.09e9 - 6.30e8i)T + (3.93e17 - 6.82e17i)T^{2} \) |
| 37 | \( 1 + (1.04e9 - 1.80e9i)T + (-3.29e18 - 5.70e18i)T^{2} \) |
| 41 | \( 1 - 1.76e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 + 3.97e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (-4.36e9 - 2.52e9i)T + (5.80e19 + 1.00e20i)T^{2} \) |
| 53 | \( 1 + (2.09e10 + 3.63e10i)T + (-2.45e20 + 4.25e20i)T^{2} \) |
| 59 | \( 1 + (2.44e10 - 1.41e10i)T + (8.89e20 - 1.54e21i)T^{2} \) |
| 61 | \( 1 + (1.97e10 + 1.14e10i)T + (1.32e21 + 2.29e21i)T^{2} \) |
| 67 | \( 1 + (2.65e10 + 4.59e10i)T + (-4.09e21 + 7.08e21i)T^{2} \) |
| 71 | \( 1 - 8.67e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-1.42e9 + 8.22e8i)T + (1.14e22 - 1.98e22i)T^{2} \) |
| 79 | \( 1 + (-1.40e11 + 2.43e11i)T + (-2.95e22 - 5.11e22i)T^{2} \) |
| 83 | \( 1 - 8.73e10iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (2.06e11 + 1.19e11i)T + (1.23e23 + 2.13e23i)T^{2} \) |
| 97 | \( 1 - 5.59e11iT - 6.93e23T^{2} \) |
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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
−19.21190869974051765432676107307, −16.57981044959840029688990299469, −15.74216925977680620866949784815, −13.04868821470413514998472469330, −11.99078391005513848618395699513, −10.82843675664646562148276402044, −8.210832495365934531928268030427, −5.01299502350381342657994524635, −3.25688386288258591703223216116, −0.16415570723003035796816210217,
4.01170704729969467262083964763, 6.47721034097130044417182462951, 7.44840423285600420008923609648, 10.73721485361107932277992750874, 12.40838784216793840699284381635, 14.46998546410670726098974854356, 15.56218943100481206932937438796, 16.80018272877569213466053554413, 18.75953176185342414053905840066, 19.96830348578199613843742935819
