L(s) = 1 | + (−0.0548 − 1.41i)2-s + (1.33 + 0.524i)3-s + (−1.99 + 0.155i)4-s + (1.33 − 0.200i)5-s + (0.667 − 1.91i)6-s + (−0.0948 + 2.64i)7-s + (0.328 + 2.80i)8-s + (−0.689 − 0.639i)9-s + (−0.356 − 1.87i)10-s + (4.38 − 4.06i)11-s + (−2.74 − 0.838i)12-s + (−0.410 − 1.80i)13-s + (3.74 − 0.0109i)14-s + (1.88 + 0.429i)15-s + (3.95 − 0.618i)16-s + (4.98 − 0.373i)17-s + ⋯ |
L(s) = 1 | + (−0.0387 − 0.999i)2-s + (0.771 + 0.302i)3-s + (−0.996 + 0.0775i)4-s + (0.595 − 0.0897i)5-s + (0.272 − 0.782i)6-s + (−0.0358 + 0.999i)7-s + (0.116 + 0.993i)8-s + (−0.229 − 0.213i)9-s + (−0.112 − 0.591i)10-s + (1.32 − 1.22i)11-s + (−0.792 − 0.241i)12-s + (−0.113 − 0.499i)13-s + (0.999 − 0.00293i)14-s + (0.486 + 0.111i)15-s + (0.987 − 0.154i)16-s + (1.20 − 0.0906i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55304 - 0.808313i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55304 - 0.808313i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0548 + 1.41i)T \) |
| 7 | \( 1 + (0.0948 - 2.64i)T \) |
good | 3 | \( 1 + (-1.33 - 0.524i)T + (2.19 + 2.04i)T^{2} \) |
| 5 | \( 1 + (-1.33 + 0.200i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (-4.38 + 4.06i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.410 + 1.80i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-4.98 + 0.373i)T + (16.8 - 2.53i)T^{2} \) |
| 19 | \( 1 + (-3.75 - 2.16i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.60 - 0.195i)T + (22.7 + 3.42i)T^{2} \) |
| 29 | \( 1 + (4.05 - 8.41i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (1.75 + 3.03i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.211 + 0.310i)T + (-13.5 + 34.4i)T^{2} \) |
| 41 | \( 1 + (1.39 - 1.11i)T + (9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.91 - 6.16i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (4.83 + 1.49i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (-1.35 + 1.99i)T + (-19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-0.602 + 3.99i)T + (-56.3 - 17.3i)T^{2} \) |
| 61 | \( 1 + (7.73 - 5.27i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (-1.76 - 3.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.76 + 5.73i)T + (-44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (1.69 + 5.49i)T + (-60.3 + 41.1i)T^{2} \) |
| 79 | \( 1 + (4.77 + 2.75i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.28 + 1.66i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (11.1 - 11.9i)T + (-6.65 - 88.7i)T^{2} \) |
| 97 | \( 1 - 0.206iT - 97T^{2} \) |
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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
−11.36691269956899115667745500432, −10.02016638061861954032226110522, −9.315303424420074670013004753030, −8.851456509084890850700378184495, −7.933703891349889897282772935354, −6.00258696190989110347247655868, −5.32220019204341742573749428699, −3.52109337573281231282545245835, −3.08591028700563192683908024981, −1.48335569594541584233942504673,
1.60269004049724909100323913845, 3.52111598822366828475628585157, 4.61099038149718114694026124082, 5.86403343663776534426706621168, 7.09238693228109132378169407682, 7.42004509992544831225354629550, 8.580818793029385924107495800912, 9.633461726468932534801963305257, 9.915715857079562512577398473312, 11.51001048683260153090229363022