| L(s) = 1 | + (−10.2 + 17.7i)2-s + (30.4 + 52.8i)3-s + (−144. − 250. i)4-s + (−133. + 232. i)5-s − 1.24e3·6-s + 3.30e3·8-s + (−765. + 1.32e3i)9-s + (−2.73e3 − 4.74e3i)10-s + (−3.42e3 − 5.93e3i)11-s + (8.83e3 − 1.53e4i)12-s + 553.·13-s − 1.63e4·15-s + (−1.52e4 + 2.64e4i)16-s + (−9.39e3 − 1.62e4i)17-s + (−1.56e4 − 2.71e4i)18-s + (−1.32e4 + 2.29e4i)19-s + ⋯ |
| L(s) = 1 | + (−0.903 + 1.56i)2-s + (0.652 + 1.12i)3-s + (−1.13 − 1.96i)4-s + (−0.479 + 0.830i)5-s − 2.35·6-s + 2.28·8-s + (−0.350 + 0.606i)9-s + (−0.865 − 1.49i)10-s + (−0.776 − 1.34i)11-s + (1.47 − 2.55i)12-s + 0.0698·13-s − 1.24·15-s + (−0.930 + 1.61i)16-s + (−0.463 − 0.802i)17-s + (−0.632 − 1.09i)18-s + (−0.443 + 0.768i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0258389 - 0.00422584i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0258389 - 0.00422584i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (10.2 - 17.7i)T + (-64 - 110. i)T^{2} \) |
| 3 | \( 1 + (-30.4 - 52.8i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (133. - 232. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 11 | \( 1 + (3.42e3 + 5.93e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 - 553.T + 6.27e7T^{2} \) |
| 17 | \( 1 + (9.39e3 + 1.62e4i)T + (-2.05e8 + 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.32e4 - 2.29e4i)T + (-4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (-7.10e3 + 1.23e4i)T + (-1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 + 2.18e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + (-8.88e4 - 1.53e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (8.33e4 - 1.44e5i)T + (-4.74e10 - 8.22e10i)T^{2} \) |
| 41 | \( 1 + 3.92e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.49e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-4.99e5 + 8.64e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 + (2.79e5 + 4.84e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.43e6 + 2.47e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-4.02e5 + 6.96e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (5.05e5 + 8.75e5i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 - 1.38e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + (-2.92e5 - 5.06e5i)T + (-5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.89e6 - 3.27e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + 3.66e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (-4.60e6 + 7.98e6i)T + (-2.21e13 - 3.83e13i)T^{2} \) |
| 97 | \( 1 + 8.21e6T + 8.07e13T^{2} \) |
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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
−15.40579491138323382651107718229, −14.63910114618427813874645732769, −13.71203851700092888583721872620, −11.00631019330939246569901621611, −10.08588764188711041143528742711, −8.893321073349030487282890295195, −8.046146071402430141360534876232, −6.71819331487059506239527469074, −5.20691687935590548849846769783, −3.39485994904492190369169001129,
0.01320323755280521912318816234, 1.49939679535123775563189655975, 2.48652988545325765918896121454, 4.33652367399940888884472520201, 7.41172828300296364631579352713, 8.293799247004782825243085211271, 9.272540044940082332469582305998, 10.65182219876307935276517950990, 12.00694088391867019795561898244, 12.83601175339037221209736752854
