L(s) = 1 | + (−0.5 + 0.866i)5-s + (−0.780 + 2.52i)7-s + (−2.39 − 4.15i)11-s + 5.94·13-s + (3.41 + 5.91i)17-s + (−4.26 + 7.39i)19-s + (4.48 − 7.77i)23-s + (−0.499 − 0.866i)25-s + 0.414·29-s + (−2.80 − 4.86i)31-s + (−1.79 − 1.93i)35-s + (−2.79 + 4.84i)37-s + 0.325·41-s + 3.90·43-s + (−4.86 + 8.43i)47-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (−0.294 + 0.955i)7-s + (−0.722 − 1.25i)11-s + 1.64·13-s + (0.828 + 1.43i)17-s + (−0.979 + 1.69i)19-s + (0.936 − 1.62i)23-s + (−0.0999 − 0.173i)25-s + 0.0770·29-s + (−0.504 − 0.873i)31-s + (−0.304 − 0.327i)35-s + (−0.460 + 0.797i)37-s + 0.0508·41-s + 0.595·43-s + (−0.710 + 1.23i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.283 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.315177691\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.315177691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.780 - 2.52i)T \) |
good | 11 | \( 1 + (2.39 + 4.15i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.94T + 13T^{2} \) |
| 17 | \( 1 + (-3.41 - 5.91i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.26 - 7.39i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.48 + 7.77i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 0.414T + 29T^{2} \) |
| 31 | \( 1 + (2.80 + 4.86i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.79 - 4.84i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 0.325T + 41T^{2} \) |
| 43 | \( 1 - 3.90T + 43T^{2} \) |
| 47 | \( 1 + (4.86 - 8.43i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.927 + 1.60i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.58 - 7.94i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.33 - 10.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.34 - 2.32i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + (-4.40 - 7.62i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.81 - 3.14i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 9.07T + 83T^{2} \) |
| 89 | \( 1 + (0.646 - 1.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.791783626095594299222209117705, −8.383865206158017291381923143615, −7.975090327529401159552333045976, −6.52528598779313585901472999555, −5.99532369342487584752079417167, −5.61458981820965478720406970815, −4.12406512887157212443094084307, −3.44024078922326161992180072146, −2.60219077953659596950827113074, −1.29442023127373395396890083953,
0.47049242538506832725394069240, 1.64678374875792205053776119784, 3.03950519460009906898456087830, 3.80675782324348140685379173611, 4.82798380875791603603968804871, 5.30014236757050083613570958557, 6.62258694518494790899406761935, 7.18734221372885859163542771443, 7.75768271705594307567651561907, 8.796220689437342980184644365494