| L(s) = 1 | + (−12.2 + 21.1i)2-s + (−79.7 − 138. i)3-s + (−43.3 − 75.1i)4-s + (1.01e3 − 1.75e3i)5-s + 3.90e3·6-s + (−4.23e3 − 4.73e3i)7-s − 1.04e4·8-s + (−2.87e3 + 4.97e3i)9-s + (2.48e4 + 4.29e4i)10-s + (−4.28e3 − 7.42e3i)11-s + (−6.91e3 + 1.19e4i)12-s − 3.84e4·13-s + (1.52e5 − 3.18e4i)14-s − 3.23e5·15-s + (1.49e5 − 2.58e5i)16-s + (1.50e5 + 2.61e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.540 + 0.936i)2-s + (−0.568 − 0.984i)3-s + (−0.0846 − 0.146i)4-s + (0.725 − 1.25i)5-s + 1.22·6-s + (−0.666 − 0.745i)7-s − 0.898·8-s + (−0.146 + 0.252i)9-s + (0.784 + 1.35i)10-s + (−0.0883 − 0.152i)11-s + (−0.0962 + 0.166i)12-s − 0.373·13-s + (1.05 − 0.221i)14-s − 1.64·15-s + (0.570 − 0.987i)16-s + (0.438 + 0.759i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.634953 - 0.483284i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.634953 - 0.483284i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (4.23e3 + 4.73e3i)T \) |
| good | 2 | \( 1 + (12.2 - 21.1i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (79.7 + 138. i)T + (-9.84e3 + 1.70e4i)T^{2} \) |
| 5 | \( 1 + (-1.01e3 + 1.75e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 11 | \( 1 + (4.28e3 + 7.42e3i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + 3.84e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + (-1.50e5 - 2.61e5i)T + (-5.92e10 + 1.02e11i)T^{2} \) |
| 19 | \( 1 + (-4.58e5 + 7.93e5i)T + (-1.61e11 - 2.79e11i)T^{2} \) |
| 23 | \( 1 + (9.66e5 - 1.67e6i)T + (-9.00e11 - 1.55e12i)T^{2} \) |
| 29 | \( 1 - 4.52e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + (1.47e5 + 2.55e5i)T + (-1.32e13 + 2.28e13i)T^{2} \) |
| 37 | \( 1 + (-8.05e6 + 1.39e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 - 1.28e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.15e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + (-1.39e7 + 2.41e7i)T + (-5.59e14 - 9.69e14i)T^{2} \) |
| 53 | \( 1 + (-2.11e7 - 3.66e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (1.17e6 + 2.03e6i)T + (-4.33e15 + 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-4.34e6 + 7.52e6i)T + (-5.84e15 - 1.01e16i)T^{2} \) |
| 67 | \( 1 + (1.26e8 + 2.18e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + 2.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + (2.32e7 + 4.02e7i)T + (-2.94e16 + 5.09e16i)T^{2} \) |
| 79 | \( 1 + (5.53e7 - 9.58e7i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.81e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (-1.45e8 + 2.52e8i)T + (-1.75e17 - 3.03e17i)T^{2} \) |
| 97 | \( 1 - 1.01e8T + 7.60e17T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−19.73502858858367431075590990251, −17.81348698580678835509297784930, −17.15962878035909553128030603726, −16.01240414646636547697776840190, −13.43205664090990919598188267585, −12.26604102842678121739279188047, −9.366355683699705990540425819681, −7.44447565576539831575856360004, −5.91543591936753400083781865257, −0.73083778939571590925591274738,
2.73196361127006012210716566321, 5.99336204531668498070537649723, 9.781142334287669662784809598023, 10.34972935049055471607693321440, 11.93098723455637644000170888389, 14.57799210037838757840881618280, 16.10283586086943409663717835286, 18.06522763285396605791794048293, 18.94775155812404732417733136898, 20.77120773034897265368017217036
