| L(s) = 1 | + (0.0439 − 1.41i)2-s + (0.723 + 2.22i)3-s + (−1.99 − 0.124i)4-s + (−1.71 − 1.43i)5-s + (3.18 − 0.925i)6-s + (−2.60 − 0.463i)7-s + (−0.263 + 2.81i)8-s + (−2.01 + 1.46i)9-s + (−2.10 + 2.36i)10-s + (2.35 − 3.24i)11-s + (−1.16 − 4.53i)12-s + (3.58 + 4.93i)13-s + (−0.769 + 3.66i)14-s + (1.95 − 4.85i)15-s + (3.96 + 0.496i)16-s + (6.90 + 2.24i)17-s + ⋯ |
| L(s) = 1 | + (0.0310 − 0.999i)2-s + (0.417 + 1.28i)3-s + (−0.998 − 0.0621i)4-s + (−0.766 − 0.641i)5-s + (1.29 − 0.377i)6-s + (−0.984 − 0.175i)7-s + (−0.0931 + 0.995i)8-s + (−0.670 + 0.487i)9-s + (−0.665 + 0.746i)10-s + (0.709 − 0.977i)11-s + (−0.337 − 1.30i)12-s + (0.995 + 1.36i)13-s + (−0.205 + 0.978i)14-s + (0.505 − 1.25i)15-s + (0.992 + 0.124i)16-s + (1.67 + 0.544i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33158 - 0.340815i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33158 - 0.340815i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0439 + 1.41i)T \) |
| 5 | \( 1 + (1.71 + 1.43i)T \) |
| 7 | \( 1 + (2.60 + 0.463i)T \) |
| good | 3 | \( 1 + (-0.723 - 2.22i)T + (-2.42 + 1.76i)T^{2} \) |
| 11 | \( 1 + (-2.35 + 3.24i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.58 - 4.93i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-6.90 - 2.24i)T + (13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.77 + 5.45i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-3.21 + 4.42i)T + (-7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.208 - 0.641i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.41 - 7.44i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.226 + 0.164i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (3.40 + 4.68i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.29iT - 43T^{2} \) |
| 47 | \( 1 + (0.315 + 0.970i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.25 - 6.94i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-4.88 + 3.55i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.920 + 1.26i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (3.72 + 1.20i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-1.74 + 0.565i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-2.82 + 3.88i)T + (-22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-6.89 + 2.24i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.78 - 5.48i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.789 + 1.08i)T + (-27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.45 - 1.77i)T + (78.4 - 57.0i)T^{2} \) |
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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
−10.46568473072967221348826708592, −9.393558209210153910916175179428, −8.974728266301347982575360412592, −8.457875155983205016544450914000, −6.84235451931488900469077272134, −5.45828311069421265770960449191, −4.42415893087875392388040137193, −3.61037633646134773274143415167, −3.26224263552711138573267817432, −1.01898364181672155873759266277,
1.04004475391392850791520536007, 3.11203112122628584957506644584, 3.76883911405025985009964715926, 5.54984903473046814493367026242, 6.31942229567354503002617101400, 7.18591301049398331754001324958, 7.70513551425003260449417388143, 8.275495782408430353040446393922, 9.573097161035926847094464065508, 10.16016147976737595729844730444
