| L(s) = 1 | + (−0.711 − 1.22i)2-s + (0.227 + 1.71i)3-s + (−0.987 + 1.73i)4-s + (−0.980 − 0.195i)5-s + (1.93 − 1.49i)6-s + (2.69 + 1.11i)7-s + (2.82 − 0.0294i)8-s + (−2.89 + 0.782i)9-s + (0.459 + 1.33i)10-s + (−4.93 + 3.29i)11-s + (−3.21 − 1.30i)12-s + (−0.992 − 4.98i)13-s + (−0.552 − 4.08i)14-s + (0.111 − 1.72i)15-s + (−2.04 − 3.43i)16-s + (3.61 + 3.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.502 − 0.864i)2-s + (0.131 + 0.991i)3-s + (−0.493 + 0.869i)4-s + (−0.438 − 0.0872i)5-s + (0.790 − 0.612i)6-s + (1.01 + 0.421i)7-s + (0.999 − 0.0103i)8-s + (−0.965 + 0.260i)9-s + (0.145 + 0.422i)10-s + (−1.48 + 0.994i)11-s + (−0.926 − 0.375i)12-s + (−0.275 − 1.38i)13-s + (−0.147 − 1.09i)14-s + (0.0288 − 0.446i)15-s + (−0.511 − 0.859i)16-s + (0.877 + 0.877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.193498 + 0.505709i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.193498 + 0.505709i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.711 + 1.22i)T \) |
| 3 | \( 1 + (-0.227 - 1.71i)T \) |
| 5 | \( 1 + (0.980 + 0.195i)T \) |
| good | 7 | \( 1 + (-2.69 - 1.11i)T + (4.94 + 4.94i)T^{2} \) |
| 11 | \( 1 + (4.93 - 3.29i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (0.992 + 4.98i)T + (-12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (-3.61 - 3.61i)T + 17iT^{2} \) |
| 19 | \( 1 + (-1.36 - 6.88i)T + (-17.5 + 7.27i)T^{2} \) |
| 23 | \( 1 + (-0.0484 + 0.0200i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (-3.42 + 5.13i)T + (-11.0 - 26.7i)T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + (6.78 + 1.35i)T + (34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (2.61 + 6.32i)T + (-28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (10.4 - 7.01i)T + (16.4 - 39.7i)T^{2} \) |
| 47 | \( 1 + (1.05 - 1.05i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.954 + 1.42i)T + (-20.2 + 48.9i)T^{2} \) |
| 59 | \( 1 + (0.480 - 2.41i)T + (-54.5 - 22.5i)T^{2} \) |
| 61 | \( 1 + (4.76 - 7.13i)T + (-23.3 - 56.3i)T^{2} \) |
| 67 | \( 1 + (-0.361 - 0.241i)T + (25.6 + 61.8i)T^{2} \) |
| 71 | \( 1 + (-0.152 + 0.367i)T + (-50.2 - 50.2i)T^{2} \) |
| 73 | \( 1 + (-2.62 - 6.32i)T + (-51.6 + 51.6i)T^{2} \) |
| 79 | \( 1 + (-0.318 + 0.318i)T - 79iT^{2} \) |
| 83 | \( 1 + (-0.150 + 0.0298i)T + (76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (-0.898 + 2.17i)T + (-62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 - 9.92iT - 97T^{2} \) |
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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
−10.24603590794848092428689698658, −9.944700408990857427482832448993, −8.639760961563260319844285837840, −7.86685174157722164177242151509, −7.81194307464964404973463142220, −5.46147376356907059859927839492, −5.06194253985675412349134709472, −3.91856491573823372033339895605, −3.02632216832581196819992770836, −1.86503284281521094678924575627,
0.28966730246448381165410992832, 1.66807160328932785861505093654, 3.13143883788246514676245358396, 4.90211433947889137736107571349, 5.29888701791537735455407654045, 6.73441783482412045484884486374, 7.20991516970176655481655331839, 7.948461815930337505573109098211, 8.551322248197612582671353139434, 9.352208424213183963533431484936
