L(s) = 1 | − 1.73i·3-s + (1.5 + 2.59i)5-s + (0.5 − 0.866i)7-s − 2.99·9-s + (−3 + 5.19i)11-s + (−1 − 1.73i)13-s + (4.5 − 2.59i)15-s + 6·17-s + 7·19-s + (−1.49 − 0.866i)21-s + (1.5 + 2.59i)23-s + (−2 + 3.46i)25-s + 5.19i·27-s + (−3 + 5.19i)29-s + (1 + 1.73i)31-s + ⋯ |
L(s) = 1 | − 0.999i·3-s + (0.670 + 1.16i)5-s + (0.188 − 0.327i)7-s − 0.999·9-s + (−0.904 + 1.56i)11-s + (−0.277 − 0.480i)13-s + (1.16 − 0.670i)15-s + 1.45·17-s + 1.60·19-s + (−0.327 − 0.188i)21-s + (0.312 + 0.541i)23-s + (−0.400 + 0.692i)25-s + 0.999i·27-s + (−0.557 + 0.964i)29-s + (0.179 + 0.311i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.716254458\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.716254458\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + (-2.5 + 4.33i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 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
−10.10988741422847538334413575735, −9.383947140078458358520778779524, −7.903983053165542946697583603814, −7.36426437142942125919744457445, −6.97158293428436286731633916590, −5.69234507888719946771370170362, −5.15389003042644954330046899002, −3.30663373907994560975155351255, −2.55325418531545072069682681972, −1.39981180678212039421183646151,
0.872297971572014747466251686579, 2.63813215317811622380824500758, 3.63935879178098036372027016555, 4.94992987742234321206390062877, 5.43571162268061002545060068987, 6.01251393840904466698967811897, 7.79846340776504668548893833396, 8.377156827410472067792029340427, 9.307061164784269253296220964657, 9.673652374575824079255213528516