| L(s) = 1 | + (0.781 − 0.623i)2-s + (−1.96 + 0.772i)3-s + (0.222 − 0.974i)4-s + (1.43 − 1.71i)5-s + (−1.05 + 1.83i)6-s + (−2.54 + 1.47i)7-s + (−0.433 − 0.900i)8-s + (1.07 − 0.999i)9-s + (0.0589 − 2.23i)10-s + (−0.994 − 4.35i)11-s + (0.315 + 2.09i)12-s + (2.72 − 3.99i)13-s + (−1.07 + 2.73i)14-s + (−1.51 + 4.47i)15-s + (−0.900 − 0.433i)16-s + (−3.91 + 0.293i)17-s + ⋯ |
| L(s) = 1 | + (0.552 − 0.440i)2-s + (−1.13 + 0.445i)3-s + (0.111 − 0.487i)4-s + (0.643 − 0.765i)5-s + (−0.431 + 0.747i)6-s + (−0.962 + 0.555i)7-s + (−0.153 − 0.318i)8-s + (0.359 − 0.333i)9-s + (0.0186 − 0.706i)10-s + (−0.299 − 1.31i)11-s + (0.0909 + 0.603i)12-s + (0.754 − 1.10i)13-s + (−0.287 + 0.731i)14-s + (−0.390 + 1.15i)15-s + (−0.225 − 0.108i)16-s + (−0.950 + 0.0712i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.368299 - 0.817722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.368299 - 0.817722i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.781 + 0.623i)T \) |
| 5 | \( 1 + (-1.43 + 1.71i)T \) |
| 43 | \( 1 + (-6.31 + 1.76i)T \) |
| good | 3 | \( 1 + (1.96 - 0.772i)T + (2.19 - 2.04i)T^{2} \) |
| 7 | \( 1 + (2.54 - 1.47i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.994 + 4.35i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-2.72 + 3.99i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (3.91 - 0.293i)T + (16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (2.66 + 2.47i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (1.37 - 4.45i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (-2.29 + 5.84i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (0.792 - 0.119i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (8.92 + 5.15i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.64 - 4.57i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (4.98 + 1.13i)T + (42.3 + 20.3i)T^{2} \) |
| 53 | \( 1 + (-4.27 - 6.27i)T + (-19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-8.78 - 4.22i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-11.5 - 1.74i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (4.34 - 4.68i)T + (-5.00 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-3.88 + 1.19i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (-4.06 + 5.95i)T + (-26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (2.33 + 4.03i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.489 + 0.192i)T + (60.8 - 56.4i)T^{2} \) |
| 89 | \( 1 + (-5.76 - 14.6i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-6.36 + 1.45i)T + (87.3 - 42.0i)T^{2} \) |
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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
−10.87993794873933254094842439551, −10.24940342045559428179564130148, −9.205289671030846215675732995983, −8.378268296003030580949030985003, −6.39624517135589877401817827925, −5.80316451439769960161955602605, −5.30361700105698676978508327809, −3.98318170802458686627112238215, −2.62276891468610318043756411254, −0.52241453084631554039652407721,
2.08377661033573447302158067858, 3.73237351443170744049102239889, 4.92184426246090449131782814637, 6.12471073147167239308245797605, 6.80283347968459622202742584794, 6.98075313811798008989032430741, 8.746803959410889168596468333755, 9.964422154433688305189345339549, 10.68355794877759142433993366179, 11.53925923398474617582442865276
