| L(s) = 1 | + (13.6 − 23.5i)2-s + (−273. − 474. i)3-s + (3.72e3 + 6.45e3i)4-s + (1.68e4 − 2.92e4i)5-s − 1.49e4·6-s + (−1.74e5 − 2.57e5i)7-s + 4.25e5·8-s + (6.47e5 − 1.12e6i)9-s + (−4.59e5 − 7.95e5i)10-s + (−4.55e6 − 7.88e6i)11-s + (2.03e6 − 3.53e6i)12-s + 2.25e7·13-s + (−8.45e6 + 6.16e5i)14-s − 1.84e7·15-s + (−2.47e7 + 4.28e7i)16-s + (2.90e7 + 5.02e7i)17-s + ⋯ |
| L(s) = 1 | + (0.150 − 0.260i)2-s + (−0.216 − 0.375i)3-s + (0.454 + 0.787i)4-s + (0.482 − 0.835i)5-s − 0.130·6-s + (−0.561 − 0.827i)7-s + 0.574·8-s + (0.405 − 0.703i)9-s + (−0.145 − 0.251i)10-s + (−0.775 − 1.34i)11-s + (0.197 − 0.341i)12-s + 1.29·13-s + (−0.299 + 0.0218i)14-s − 0.418·15-s + (−0.368 + 0.638i)16-s + (0.291 + 0.505i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.40459 - 1.22528i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.40459 - 1.22528i\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.74e5 + 2.57e5i)T \) |
| good | 2 | \( 1 + (-13.6 + 23.5i)T + (-4.09e3 - 7.09e3i)T^{2} \) |
| 3 | \( 1 + (273. + 474. i)T + (-7.97e5 + 1.38e6i)T^{2} \) |
| 5 | \( 1 + (-1.68e4 + 2.92e4i)T + (-6.10e8 - 1.05e9i)T^{2} \) |
| 11 | \( 1 + (4.55e6 + 7.88e6i)T + (-1.72e13 + 2.98e13i)T^{2} \) |
| 13 | \( 1 - 2.25e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + (-2.90e7 - 5.02e7i)T + (-4.95e15 + 8.57e15i)T^{2} \) |
| 19 | \( 1 + (1.30e8 - 2.25e8i)T + (-2.10e16 - 3.64e16i)T^{2} \) |
| 23 | \( 1 + (-4.93e8 + 8.55e8i)T + (-2.52e17 - 4.36e17i)T^{2} \) |
| 29 | \( 1 - 3.04e8T + 1.02e19T^{2} \) |
| 31 | \( 1 + (-5.06e8 - 8.77e8i)T + (-1.22e19 + 2.11e19i)T^{2} \) |
| 37 | \( 1 + (5.24e9 - 9.07e9i)T + (-1.21e20 - 2.10e20i)T^{2} \) |
| 41 | \( 1 + 8.80e9T + 9.25e20T^{2} \) |
| 43 | \( 1 - 1.24e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (1.94e10 - 3.36e10i)T + (-2.73e21 - 4.72e21i)T^{2} \) |
| 53 | \( 1 + (-2.30e10 - 3.99e10i)T + (-1.30e22 + 2.25e22i)T^{2} \) |
| 59 | \( 1 + (-2.82e11 - 4.89e11i)T + (-5.24e22 + 9.09e22i)T^{2} \) |
| 61 | \( 1 + (-3.51e11 + 6.08e11i)T + (-8.09e22 - 1.40e23i)T^{2} \) |
| 67 | \( 1 + (3.16e11 + 5.47e11i)T + (-2.74e23 + 4.74e23i)T^{2} \) |
| 71 | \( 1 - 1.25e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + (-8.27e11 - 1.43e12i)T + (-8.35e23 + 1.44e24i)T^{2} \) |
| 79 | \( 1 + (-1.19e12 + 2.07e12i)T + (-2.33e24 - 4.04e24i)T^{2} \) |
| 83 | \( 1 - 8.40e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + (-1.16e12 + 2.01e12i)T + (-1.09e25 - 1.90e25i)T^{2} \) |
| 97 | \( 1 + 7.28e12T + 6.73e25T^{2} \) |
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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
−18.76316864973351844862670942431, −17.00352686883674342136274356026, −16.15495940163869025266143163637, −13.39736375522612228345842433192, −12.62049516034112976744881404308, −10.72140003115610196005196098720, −8.386636223612333715537693176580, −6.33650073416726159071005581856, −3.61026520961460632666900944877, −1.05162068651292958410399294702,
2.25568912853358882026861363893, 5.30094980453390818294671175153, 6.88628272990497838907145156105, 9.817961277853320470563350404905, 10.98106091058687139010970724664, 13.35257477986194946007273934149, 15.15183006559431195362014093348, 15.91564359244608786807724035807, 18.09737753110994795284936914649, 19.25493771092016381725741032946
