L(s) = 1 | + (−0.978 − 0.207i)2-s + (1.62 + 0.597i)3-s + (0.913 + 0.406i)4-s + (0.605 − 2.15i)5-s + (−1.46 − 0.922i)6-s + (−0.294 + 0.510i)7-s + (−0.809 − 0.587i)8-s + (2.28 + 1.94i)9-s + (−1.03 + 1.97i)10-s + (0.140 + 0.0299i)11-s + (1.24 + 1.20i)12-s + (4.09 − 0.871i)13-s + (0.394 − 0.438i)14-s + (2.27 − 3.13i)15-s + (0.669 + 0.743i)16-s + (−1.93 − 1.40i)17-s + ⋯ |
L(s) = 1 | + (−0.691 − 0.147i)2-s + (0.938 + 0.344i)3-s + (0.456 + 0.203i)4-s + (0.270 − 0.962i)5-s + (−0.598 − 0.376i)6-s + (−0.111 + 0.193i)7-s + (−0.286 − 0.207i)8-s + (0.762 + 0.647i)9-s + (−0.328 + 0.625i)10-s + (0.0424 + 0.00903i)11-s + (0.358 + 0.348i)12-s + (1.13 − 0.241i)13-s + (0.105 − 0.117i)14-s + (0.586 − 0.810i)15-s + (0.167 + 0.185i)16-s + (−0.470 − 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49853 - 0.239678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49853 - 0.239678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (-1.62 - 0.597i)T \) |
| 5 | \( 1 + (-0.605 + 2.15i)T \) |
good | 7 | \( 1 + (0.294 - 0.510i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.140 - 0.0299i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-4.09 + 0.871i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (1.93 + 1.40i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.37 - 0.996i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.81 + 4.23i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.195 + 1.86i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.586 - 5.58i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.528 + 1.62i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.630 - 0.134i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (3.45 - 5.98i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.564 + 5.36i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-3.80 + 2.76i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-7.22 + 1.53i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (4.51 + 0.958i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (0.357 - 3.40i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (12.7 - 9.25i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (4.82 + 14.8i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.00170 + 0.0162i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (10.2 - 4.58i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-3.02 - 9.31i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.981 + 9.34i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−10.76359282868048362994638070210, −9.967392781928836098423080567883, −8.973793813371023714873608167445, −8.695912326042016655832573856214, −7.77030445681759910500725558184, −6.54526132807749808019246876030, −5.21546179634971701347554748429, −4.02122375821466487912177716385, −2.74886640898738597137553267694, −1.34757924134908175120866410443,
1.57498629256679929510682161715, 2.86990554223945246840410999595, 3.89113675256068985950496749933, 5.86155415681974156178202200470, 6.82667591943524902702561483352, 7.42425938672703315204389969086, 8.497014367434778699649343359945, 9.233464327285240828139439342949, 10.09567723002474780492648033397, 10.95674193449494214505358785161