L(s) = 1 | + (0.258 − 0.965i)2-s + (−1.28 + 1.16i)3-s + (−0.866 − 0.499i)4-s + (0.946 − 2.02i)5-s + (0.789 + 1.54i)6-s + (−1.09 − 2.40i)7-s + (−0.707 + 0.707i)8-s + (0.302 − 2.98i)9-s + (−1.71 − 1.43i)10-s + (1.34 + 2.32i)11-s + (1.69 − 0.363i)12-s + (−6.36 + 1.70i)13-s + (−2.60 + 0.436i)14-s + (1.13 + 3.70i)15-s + (0.500 + 0.866i)16-s + (−1.16 + 1.16i)17-s + ⋯ |
L(s) = 1 | + (0.183 − 0.683i)2-s + (−0.741 + 0.670i)3-s + (−0.433 − 0.249i)4-s + (0.423 − 0.906i)5-s + (0.322 + 0.629i)6-s + (−0.414 − 0.909i)7-s + (−0.249 + 0.249i)8-s + (0.100 − 0.994i)9-s + (−0.541 − 0.454i)10-s + (0.405 + 0.701i)11-s + (0.488 − 0.104i)12-s + (−1.76 + 0.473i)13-s + (−0.697 + 0.116i)14-s + (0.293 + 0.955i)15-s + (0.125 + 0.216i)16-s + (−0.283 + 0.283i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0389878 + 0.463541i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0389878 + 0.463541i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 3 | \( 1 + (1.28 - 1.16i)T \) |
| 5 | \( 1 + (-0.946 + 2.02i)T \) |
| 7 | \( 1 + (1.09 + 2.40i)T \) |
good | 11 | \( 1 + (-1.34 - 2.32i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (6.36 - 1.70i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (1.16 - 1.16i)T - 17iT^{2} \) |
| 19 | \( 1 - 2.59T + 19T^{2} \) |
| 23 | \( 1 + (1.94 + 7.27i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (8.02 - 4.63i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (1.01 + 0.583i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.346 + 0.346i)T + 37iT^{2} \) |
| 41 | \( 1 + (8.19 + 4.73i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.101 + 0.0271i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (6.41 + 1.71i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (5.41 - 5.41i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.290 - 0.503i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.4 + 6.62i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.62 + 2.31i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 5.20T + 71T^{2} \) |
| 73 | \( 1 + (6.71 + 6.71i)T + 73iT^{2} \) |
| 79 | \( 1 + (-2.22 + 1.28i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.89 + 10.8i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 1.13T + 89T^{2} \) |
| 97 | \( 1 + (-8.78 - 2.35i)T + (84.0 + 48.5i)T^{2} \) |
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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
−9.966044847189718492915111516665, −9.721945429088016521696908775567, −8.846578724541960285444107964997, −7.32284992903061560377032180601, −6.39796820736258703226809158205, −5.09018644189292667960873103474, −4.62220482516440647830352781619, −3.66893467331115055501025068281, −1.91015027516717461547923297362, −0.24738648302552577863816162055,
2.17554487285203885227676619468, 3.35216014918047114354126644093, 5.20299900246044970808569243939, 5.66513550412129539670990419661, 6.57796475389735174654145487231, 7.28971366600324661290267667487, 8.079682536546051261628554450865, 9.511915077752687040217435304383, 9.942689598733146392375330590197, 11.44301854364490767925418550258