| L(s) = 1 | + (0.851 − 0.523i)2-s + (0.0903 − 0.241i)3-s + (0.451 − 0.892i)4-s + (−1.19 − 1.62i)5-s + (−0.0493 − 0.252i)6-s + (1.15 − 1.76i)7-s + (−0.0825 − 0.996i)8-s + (2.21 + 1.92i)9-s + (−1.86 − 0.759i)10-s + (0.499 + 0.171i)11-s + (−0.174 − 0.189i)12-s + (−1.18 − 1.44i)13-s + (0.0579 − 2.10i)14-s + (−0.500 + 0.141i)15-s + (−0.592 − 0.805i)16-s + (−2.08 − 4.12i)17-s + ⋯ |
| L(s) = 1 | + (0.602 − 0.370i)2-s + (0.0521 − 0.139i)3-s + (0.225 − 0.446i)4-s + (−0.534 − 0.727i)5-s + (−0.0201 − 0.103i)6-s + (0.434 − 0.665i)7-s + (−0.0291 − 0.352i)8-s + (0.737 + 0.642i)9-s + (−0.591 − 0.240i)10-s + (0.150 + 0.0517i)11-s + (−0.0503 − 0.0547i)12-s + (−0.330 − 0.400i)13-s + (0.0154 − 0.561i)14-s + (−0.129 + 0.0365i)15-s + (−0.148 − 0.201i)16-s + (−0.506 − 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19284 - 1.64012i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19284 - 1.64012i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.851 + 0.523i)T \) |
| 19 | \( 1 + (-3.40 - 2.72i)T \) |
| good | 3 | \( 1 + (-0.0903 + 0.241i)T + (-2.26 - 1.97i)T^{2} \) |
| 5 | \( 1 + (1.19 + 1.62i)T + (-1.49 + 4.77i)T^{2} \) |
| 7 | \( 1 + (-1.15 + 1.76i)T + (-2.81 - 6.41i)T^{2} \) |
| 11 | \( 1 + (-0.499 - 0.171i)T + (8.68 + 6.75i)T^{2} \) |
| 13 | \( 1 + (1.18 + 1.44i)T + (-2.49 + 12.7i)T^{2} \) |
| 17 | \( 1 + (2.08 + 4.12i)T + (-10.0 + 13.6i)T^{2} \) |
| 23 | \( 1 + (2.63 + 7.04i)T + (-17.3 + 15.1i)T^{2} \) |
| 29 | \( 1 + (1.03 + 0.0568i)T + (28.8 + 3.19i)T^{2} \) |
| 31 | \( 1 + (7.62 - 4.12i)T + (16.9 - 25.9i)T^{2} \) |
| 37 | \( 1 + (-2.09 - 0.720i)T + (29.1 + 22.7i)T^{2} \) |
| 41 | \( 1 + (-4.91 - 4.78i)T + (1.12 + 40.9i)T^{2} \) |
| 43 | \( 1 + (0.326 - 0.132i)T + (30.8 - 29.9i)T^{2} \) |
| 47 | \( 1 + (1.18 + 6.07i)T + (-43.5 + 17.6i)T^{2} \) |
| 53 | \( 1 + (1.16 + 5.96i)T + (-49.1 + 19.9i)T^{2} \) |
| 59 | \( 1 + (-5.71 - 5.55i)T + (1.62 + 58.9i)T^{2} \) |
| 61 | \( 1 + (4.51 - 2.12i)T + (38.7 - 47.0i)T^{2} \) |
| 67 | \( 1 + (-7.78 + 5.40i)T + (23.4 - 62.7i)T^{2} \) |
| 71 | \( 1 + (-11.3 - 5.33i)T + (45.1 + 54.8i)T^{2} \) |
| 73 | \( 1 + (-2.81 - 5.56i)T + (-43.2 + 58.8i)T^{2} \) |
| 79 | \( 1 + (-8.18 + 3.32i)T + (56.6 - 55.0i)T^{2} \) |
| 83 | \( 1 + (-2.22 - 5.07i)T + (-56.2 + 61.0i)T^{2} \) |
| 89 | \( 1 + (1.30 - 2.58i)T + (-52.7 - 71.7i)T^{2} \) |
| 97 | \( 1 + (1.61 + 1.11i)T + (34.0 + 90.8i)T^{2} \) |
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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
−10.33167248455538670660757370550, −9.445420092788060742779489353621, −8.259947641691163499614204739704, −7.55481078736955459657245412342, −6.70312457033178490746678468641, −5.23782981120213684972685161816, −4.62760843940578768158962835452, −3.80288583979551459292204956472, −2.29592946164817947625023461424, −0.910061200960857433318841061133,
1.98219488091610857064685891283, 3.42120261993230933256258195380, 4.10599580076842754017122594926, 5.29341321252388929297990456161, 6.23901878171375588047805278961, 7.20138194532760156351793532335, 7.74570538615703793349716363011, 8.998528851173959254840055885839, 9.657318144163174938784943715236, 11.00486737398178669631498600908
