| L(s) = 1 | + (−0.642 + 0.136i)2-s + (−0.199 − 1.90i)3-s + (−1.43 + 0.638i)4-s + (−2.38 + 2.65i)5-s + (0.388 + 1.19i)6-s + (−2.59 + 0.531i)7-s + (1.89 − 1.37i)8-s + (−0.642 + 0.136i)9-s + (1.17 − 2.02i)10-s + (1.5 + 2.59i)12-s + (−1.82 + 5.62i)13-s + (1.59 − 0.695i)14-s + (5.52 + 4.01i)15-s + (1.06 − 1.18i)16-s + (1.62 + 0.344i)17-s + (0.393 − 0.175i)18-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.0965i)2-s + (−0.115 − 1.09i)3-s + (−0.716 + 0.319i)4-s + (−1.06 + 1.18i)5-s + (0.158 + 0.487i)6-s + (−0.979 + 0.200i)7-s + (0.670 − 0.486i)8-s + (−0.214 + 0.0455i)9-s + (0.370 − 0.641i)10-s + (0.433 + 0.749i)12-s + (−0.506 + 1.55i)13-s + (0.425 − 0.185i)14-s + (1.42 + 1.03i)15-s + (0.267 − 0.297i)16-s + (0.393 + 0.0835i)17-s + (0.0928 − 0.0413i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0912 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0912 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.258721 - 0.236107i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.258721 - 0.236107i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.59 - 0.531i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (0.642 - 0.136i)T + (1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (0.199 + 1.90i)T + (-2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (2.38 - 2.65i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (1.82 - 5.62i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.62 - 0.344i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-1.35 - 0.602i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (1.67 + 2.89i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.49 + 1.81i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (4.73 + 5.26i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.471 + 4.48i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-1.04 + 0.755i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 1.59T + 43T^{2} \) |
| 47 | \( 1 + (1.51 + 0.673i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-6.17 - 6.85i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-8.08 + 3.60i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (4.47 - 4.96i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.91 + 8.50i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (2.66 + 8.19i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.16 + 1.85i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-6.25 + 1.32i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-0.0518 - 0.159i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.28 - 2.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.00 + 9.26i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−9.817111687046147185232517906737, −9.168613537429433531795325848384, −8.026895534633920335043793027474, −7.31953692558968492168794451567, −6.95232328640541492186312040904, −6.02081901696263062166481774513, −4.26953121455921800562547706633, −3.57015393308738938638238263100, −2.23692338220261857570668012108, −0.29016721050179524851902103059,
0.900286368683628097123299368153, 3.39464665564063080126792803356, 4.04815204668885961372628233063, 5.10262991277332219371760701635, 5.45446533327894104847276811497, 7.30347042926297850368985801840, 8.093907485920224313142264940936, 8.899017539662824875059346039714, 9.629244929502327386965749865381, 10.12954455717079996798978788957
