L(s) = 1 | + (1.22 − 0.707i)2-s − 2.44·5-s + (2.5 + 0.866i)7-s + 2.82i·8-s + (−2.99 + 1.73i)10-s + 1.41i·11-s + (4.5 − 2.59i)13-s + (3.67 − 0.707i)14-s + (2.00 + 3.46i)16-s + (2.44 + 4.24i)17-s + (1.5 + 0.866i)19-s + (1.00 + 1.73i)22-s + 5.65i·23-s + 0.999·25-s + (3.67 − 6.36i)26-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)2-s − 1.09·5-s + (0.944 + 0.327i)7-s + 0.999i·8-s + (−0.948 + 0.547i)10-s + 0.426i·11-s + (1.24 − 0.720i)13-s + (0.981 − 0.188i)14-s + (0.500 + 0.866i)16-s + (0.594 + 1.02i)17-s + (0.344 + 0.198i)19-s + (0.213 + 0.369i)22-s + 1.17i·23-s + 0.199·25-s + (0.720 − 1.24i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.97103 + 0.412328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97103 + 0.412328i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.22 + 0.707i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 - 1.41iT - 11T^{2} \) |
| 13 | \( 1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.44 - 4.24i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.5 - 0.866i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.65iT - 23T^{2} \) |
| 29 | \( 1 + (-2.44 - 1.41i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.67 + 6.36i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.12 + 10.6i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.44 - 1.41i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.44 + 4.24i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 1.73i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.07iT - 71T^{2} \) |
| 73 | \( 1 + (-1.5 + 0.866i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.67 + 6.36i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.44 - 4.24i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9 + 5.19i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.16891789193882987488113909891, −10.30160929331047313107524908295, −8.727985839479846086002192914075, −8.150201953600577860913937933968, −7.48422665484770251711252216365, −5.83488728820976416365292429234, −5.04911992710713684753711530551, −3.90868214594542927641564063072, −3.39229902308404054209432818102, −1.71545602195706601283614000430,
1.02025702418944601500794049073, 3.28164616695823343451397348105, 4.27078655163865553323861757996, 4.87631840434462117101426677532, 6.06503562516872818945650245285, 6.98575851100083715731601629340, 7.893871690998630528431346819007, 8.680892339168043670670756714350, 9.841986075784309960037952087325, 11.03088147010024483327009036040