| L(s) = 1 | + (−0.317 − 0.747i)2-s + (−2.54 + 0.493i)3-s + (0.930 − 0.964i)4-s + (−1.28 − 1.82i)5-s + (1.17 + 1.74i)6-s + (0.700 − 2.61i)7-s + (−2.53 − 0.972i)8-s + (3.42 − 1.38i)9-s + (−0.956 + 1.54i)10-s + (2.63 + 0.559i)11-s + (−1.88 + 2.90i)12-s + (−2.73 + 0.336i)13-s + (−2.17 + 0.306i)14-s + (4.17 + 4.00i)15-s + (−0.0166 − 0.476i)16-s + (1.05 − 1.75i)17-s + ⋯ |
| L(s) = 1 | + (−0.224 − 0.528i)2-s + (−1.46 + 0.285i)3-s + (0.465 − 0.482i)4-s + (−0.576 − 0.816i)5-s + (0.479 + 0.711i)6-s + (0.264 − 0.988i)7-s + (−0.895 − 0.343i)8-s + (1.14 − 0.461i)9-s + (−0.302 + 0.488i)10-s + (0.793 + 0.168i)11-s + (−0.545 + 0.839i)12-s + (−0.759 + 0.0932i)13-s + (−0.582 + 0.0818i)14-s + (1.07 + 1.03i)15-s + (−0.00415 − 0.119i)16-s + (0.255 − 0.424i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.107521 + 0.367309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.107521 + 0.367309i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.28 + 1.82i)T \) |
| 19 | \( 1 + (1.43 + 4.11i)T \) |
| good | 2 | \( 1 + (0.317 + 0.747i)T + (-1.38 + 1.43i)T^{2} \) |
| 3 | \( 1 + (2.54 - 0.493i)T + (2.78 - 1.12i)T^{2} \) |
| 7 | \( 1 + (-0.700 + 2.61i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.63 - 0.559i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (2.73 - 0.336i)T + (12.6 - 3.14i)T^{2} \) |
| 17 | \( 1 + (-1.05 + 1.75i)T + (-7.98 - 15.0i)T^{2} \) |
| 23 | \( 1 + (1.79 - 5.88i)T + (-19.0 - 12.8i)T^{2} \) |
| 29 | \( 1 + (1.49 - 5.99i)T + (-25.6 - 13.6i)T^{2} \) |
| 31 | \( 1 + (5.90 + 0.620i)T + (30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (1.66 + 3.27i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (1.98 - 0.0694i)T + (40.9 - 2.86i)T^{2} \) |
| 43 | \( 1 + (3.16 + 4.51i)T + (-14.7 + 40.4i)T^{2} \) |
| 47 | \( 1 + (1.26 - 0.760i)T + (22.0 - 41.4i)T^{2} \) |
| 53 | \( 1 + (-0.100 - 5.77i)T + (-52.9 + 1.84i)T^{2} \) |
| 59 | \( 1 + (4.85 - 3.03i)T + (25.8 - 53.0i)T^{2} \) |
| 61 | \( 1 + (-1.88 + 1.00i)T + (34.1 - 50.5i)T^{2} \) |
| 67 | \( 1 + (-3.78 + 4.35i)T + (-9.32 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-6.23 + 3.04i)T + (43.7 - 55.9i)T^{2} \) |
| 73 | \( 1 + (-12.6 - 1.55i)T + (70.8 + 17.6i)T^{2} \) |
| 79 | \( 1 + (-6.90 + 10.2i)T + (-29.5 - 73.2i)T^{2} \) |
| 83 | \( 1 + (3.24 + 4.00i)T + (-17.2 + 81.1i)T^{2} \) |
| 89 | \( 1 + (-0.0512 + 1.46i)T + (-88.7 - 6.20i)T^{2} \) |
| 97 | \( 1 + (8.06 - 7.01i)T + (13.4 - 96.0i)T^{2} \) |
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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
−10.85485361929331966331683579067, −9.786326751452074425256998754329, −9.102735717856737816100226122567, −7.42667065310032917984185167282, −6.80597324624219854796056103413, −5.53603681645701063439077924094, −4.82817281298818804547976286937, −3.72799100979201978416577014681, −1.49274661053511198402447151710, −0.30341975879259422860085081226,
2.26743789562464917780523387553, 3.79411839538383272911924846230, 5.30612969861678238688398862627, 6.28984215434146381539978030543, 6.64557461461826778887883931441, 7.76365112737468010103486925593, 8.498591487571205896699240353309, 9.927176301933018555058442920269, 10.98486878090723671704730015155, 11.59067178912349571019010318731
