| L(s) = 1 | + (0.173 + 0.984i)2-s + (1.17 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−1.87 − 0.684i)5-s + (1.17 + 0.984i)6-s + (−1.34 + 2.33i)7-s + (−0.5 − 0.866i)8-s + (−0.113 + 0.642i)9-s + (0.347 − 1.96i)10-s + (−1.59 − 2.75i)11-s + (−0.766 + 1.32i)12-s + (−4.41 − 3.70i)13-s + (−2.53 − 0.921i)14-s + (−2.87 + 1.04i)15-s + (0.766 − 0.642i)16-s + (−1.13 − 6.41i)17-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (0.677 − 0.568i)3-s + (−0.469 + 0.171i)4-s + (−0.840 − 0.305i)5-s + (0.479 + 0.402i)6-s + (−0.509 + 0.882i)7-s + (−0.176 − 0.306i)8-s + (−0.0377 + 0.214i)9-s + (0.109 − 0.622i)10-s + (−0.480 − 0.831i)11-s + (−0.221 + 0.383i)12-s + (−1.22 − 1.02i)13-s + (−0.676 − 0.246i)14-s + (−0.743 + 0.270i)15-s + (0.191 − 0.160i)16-s + (−0.274 − 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.194697 - 0.369499i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.194697 - 0.369499i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-1.17 + 0.984i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (1.87 + 0.684i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (1.34 - 2.33i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (4.41 + 3.70i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.13 + 6.41i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.652 + 0.237i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.490 - 2.78i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (1.22 - 2.12i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.36T + 37T^{2} \) |
| 41 | \( 1 + (0.266 - 0.223i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (5.69 + 2.07i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.36 + 7.76i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (7.71 - 2.80i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.0996 - 0.565i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.75 - 1.00i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (0.860 - 4.88i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.94 - 2.89i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-12.0 + 10.1i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (6.94 - 5.82i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (4.23 - 7.34i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.92 - 4.97i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (0.0603 + 0.342i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−9.831771241216541821296913485214, −8.880179098025433695584760708582, −8.269086585269914639551134344367, −7.57323008712722479399771266845, −6.87366957717713393729129050236, −5.47550685527546180752164818441, −4.94323859567521483493931398414, −3.29147578667384449464571339826, −2.57175993283454399852336348925, −0.17872596496707226183241670003,
2.07860501360803805221682047096, 3.36535679394690600630186610188, 4.03151363832239569109410712648, 4.71837015058916641015303025900, 6.43001080515062826735042548991, 7.36574566203942824724041137587, 8.186936008590147552408669324745, 9.318023279479073764625880698208, 9.875566951213717638982368931800, 10.55328887192095007087624791511
