| L(s) = 1 | + (−4.25e3 + 7.37e3i)2-s + (−1.76e5 − 3.06e5i)3-s + (−1.94e7 − 3.36e7i)4-s + (−3.85e8 + 6.67e8i)5-s + 3.00e9·6-s + (3.54e10 − 9.11e9i)7-s + (4.54e10 + 3.05e−5i)8-s + (3.61e11 − 6.25e11i)9-s + (−3.27e12 − 5.68e12i)10-s + (8.18e12 + 1.41e13i)11-s + (−6.87e12 + 1.19e13i)12-s + 1.04e14·13-s + (−8.37e13 + 3.00e14i)14-s + 2.72e14·15-s + (4.59e14 − 7.95e14i)16-s + (1.52e15 + 2.63e15i)17-s + ⋯ |
| L(s) = 1 | + (−0.734 + 1.27i)2-s + (−0.191 − 0.332i)3-s + (−0.579 − 1.00i)4-s + (−0.705 + 1.22i)5-s + 0.564·6-s + (0.968 − 0.248i)7-s + 0.233·8-s + (0.426 − 0.738i)9-s + (−1.03 − 1.79i)10-s + (0.786 + 1.36i)11-s + (−0.222 + 0.385i)12-s + 1.24·13-s + (−0.394 + 1.41i)14-s + 0.542·15-s + (0.407 − 0.706i)16-s + (0.633 + 1.09i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.207i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(\approx\) |
\(1.197394676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197394676\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-3.54e10 + 9.11e9i)T \) |
| good | 2 | \( 1 + (4.25e3 - 7.37e3i)T + (-1.67e7 - 2.90e7i)T^{2} \) |
| 3 | \( 1 + (1.76e5 + 3.06e5i)T + (-4.23e11 + 7.33e11i)T^{2} \) |
| 5 | \( 1 + (3.85e8 - 6.67e8i)T + (-1.49e17 - 2.58e17i)T^{2} \) |
| 11 | \( 1 + (-8.18e12 - 1.41e13i)T + (-5.41e25 + 9.38e25i)T^{2} \) |
| 13 | \( 1 - 1.04e14T + 7.05e27T^{2} \) |
| 17 | \( 1 + (-1.52e15 - 2.63e15i)T + (-2.88e30 + 4.99e30i)T^{2} \) |
| 19 | \( 1 + (5.48e15 - 9.50e15i)T + (-4.65e31 - 8.06e31i)T^{2} \) |
| 23 | \( 1 + (-8.92e16 + 1.54e17i)T + (-5.52e33 - 9.56e33i)T^{2} \) |
| 29 | \( 1 - 2.60e16T + 3.63e36T^{2} \) |
| 31 | \( 1 + (9.31e16 + 1.61e17i)T + (-9.61e36 + 1.66e37i)T^{2} \) |
| 37 | \( 1 + (6.36e18 - 1.10e19i)T + (-8.01e38 - 1.38e39i)T^{2} \) |
| 41 | \( 1 + 8.72e19T + 2.08e40T^{2} \) |
| 43 | \( 1 + 2.81e20T + 6.86e40T^{2} \) |
| 47 | \( 1 + (-4.37e19 + 7.57e19i)T + (-3.17e41 - 5.49e41i)T^{2} \) |
| 53 | \( 1 + (-1.64e21 - 2.85e21i)T + (-6.39e42 + 1.10e43i)T^{2} \) |
| 59 | \( 1 + (-7.48e21 - 1.29e22i)T + (-9.33e43 + 1.61e44i)T^{2} \) |
| 61 | \( 1 + (8.55e21 - 1.48e22i)T + (-2.14e44 - 3.72e44i)T^{2} \) |
| 67 | \( 1 + (-3.29e22 - 5.71e22i)T + (-2.24e45 + 3.88e45i)T^{2} \) |
| 71 | \( 1 - 1.42e22T + 1.91e46T^{2} \) |
| 73 | \( 1 + (4.75e22 + 8.24e22i)T + (-1.91e46 + 3.31e46i)T^{2} \) |
| 79 | \( 1 + (1.41e22 - 2.45e22i)T + (-1.37e47 - 2.38e47i)T^{2} \) |
| 83 | \( 1 - 1.18e24T + 9.48e47T^{2} \) |
| 89 | \( 1 + (1.53e24 - 2.66e24i)T + (-2.71e48 - 4.70e48i)T^{2} \) |
| 97 | \( 1 - 9.74e23T + 4.66e49T^{2} \) |
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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
−16.92786784724616459194172596399, −15.01288495697757409833986419320, −14.78111660985093632199291407133, −12.08286930056057033083220108041, −10.40616872293731550092772761046, −8.413674217435882095671415151631, −7.17464320132360356359781222211, −6.35509494454057113345551276307, −3.92452187492969150523568458525, −1.29484931679808271749851113118,
0.63057449998706328493148546176, 1.48674180527935401049186708120, 3.59107318093756878080650555320, 5.11976097158325997557860414244, 8.248127256366598811557733858500, 9.114424746667278474648022064909, 11.10809621144063476304572507606, 11.64114035454498129430507416015, 13.38018740450160172879256350096, 15.78576643528268340349506729188
