L(s) = 1 | + (−0.5 + 0.866i)5-s + (2 + 3.46i)11-s + (−3 + 5.19i)13-s − 6·17-s − 4·19-s + (−0.499 − 0.866i)25-s + (1 + 1.73i)29-s + (4 − 6.92i)31-s − 2·37-s + (3 − 5.19i)41-s + (−6 − 10.3i)43-s + (−4 − 6.92i)47-s + (3.5 − 6.06i)49-s + 6·53-s − 3.99·55-s + ⋯ |
L(s) = 1 | + (−0.223 + 0.387i)5-s + (0.603 + 1.04i)11-s + (−0.832 + 1.44i)13-s − 1.45·17-s − 0.917·19-s + (−0.0999 − 0.173i)25-s + (0.185 + 0.321i)29-s + (0.718 − 1.24i)31-s − 0.328·37-s + (0.468 − 0.811i)41-s + (−0.914 − 1.58i)43-s + (−0.583 − 1.01i)47-s + (0.5 − 0.866i)49-s + 0.824·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
good | 7 | \( 1 + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (3 - 5.19i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1 - 1.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + (-3 + 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6 + 10.3i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (6 - 10.3i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7 + 12.1i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 - 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.499466376583209059613860942847, −7.40600431903669641257903946210, −6.78629005098767373322008202528, −6.48506926316086115192751231418, −5.14889466791944335456037666305, −4.31585622469703807855010308937, −3.91663436435070860240344292739, −2.35199065601340311288318228898, −1.95014402850216305125437303690, 0,
1.19723752386440659352610619767, 2.58676221891270197606603675293, 3.32764672861965829004093677583, 4.45905685587002289731098115620, 4.95848890397936305559578185465, 6.10449497952573634087183106523, 6.50321884540174078969104703187, 7.62911776757967538617663112329, 8.242155141583757194759864805876