L(s) = 1 | + 3.16·5-s + (2 − 1.73i)7-s + 6.32·11-s + (1.58 − 2.73i)17-s + (3.5 + 6.06i)19-s − 3.16·23-s + 5.00·25-s + (−1.58 − 2.73i)29-s + (−1.5 − 2.59i)31-s + (6.32 − 5.47i)35-s + (2 + 3.46i)37-s + (−4.74 + 8.21i)41-s + (−2.5 − 4.33i)43-s + (−4.74 + 8.21i)47-s + (1.00 − 6.92i)49-s + ⋯ |
L(s) = 1 | + 1.41·5-s + (0.755 − 0.654i)7-s + 1.90·11-s + (0.383 − 0.664i)17-s + (0.802 + 1.39i)19-s − 0.659·23-s + 1.00·25-s + (−0.293 − 0.508i)29-s + (−0.269 − 0.466i)31-s + (1.06 − 0.925i)35-s + (0.328 + 0.569i)37-s + (−0.740 + 1.28i)41-s + (−0.381 − 0.660i)43-s + (−0.691 + 1.19i)47-s + (0.142 − 0.989i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.955669801\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.955669801\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
good | 5 | \( 1 - 3.16T + 5T^{2} \) |
| 11 | \( 1 - 6.32T + 11T^{2} \) |
| 13 | \( 1 + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.58 + 2.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.5 - 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 3.16T + 23T^{2} \) |
| 29 | \( 1 + (1.58 + 2.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.74 - 8.21i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4.74 - 8.21i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.74 - 8.21i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.32 + 10.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 2.59i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + (-2.5 + 4.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.16 + 5.47i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.74 - 8.21i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.296218892531216413651611358616, −8.172870215742775303835731822175, −7.49429087573113733680634317944, −6.42719822058545440012541027781, −6.03495650269185642764150361312, −5.05055661837180133185904415108, −4.17165018108526593490699870056, −3.23212065675768586009343371977, −1.73962862597391452885871322123, −1.32185881463751577042186789133,
1.37792663173209832953096329668, 1.94421461365175977286967152565, 3.15877771887804722088392555099, 4.28886935271979262304470830058, 5.26936470622370529564700347626, 5.86040659014193410100160977440, 6.61025892505633417959271883855, 7.36845634970292825448500174187, 8.708017163381552161163134492903, 8.928788288142498036402677624918