L(s) = 1 | + (−0.928 − 0.371i)2-s + (0.654 − 0.755i)3-s + (0.723 + 0.690i)4-s + (−0.841 + 0.540i)5-s + (−0.888 + 0.458i)6-s + (0.786 − 0.618i)7-s + (−0.415 − 0.909i)8-s + (−0.142 − 0.989i)9-s + (0.981 − 0.189i)10-s + (0.888 + 0.458i)11-s + (0.995 − 0.0950i)12-s + (−0.580 − 0.814i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (0.0475 + 0.998i)16-s + (0.723 − 0.690i)17-s + ⋯ |
L(s) = 1 | + (−0.928 − 0.371i)2-s + (0.654 − 0.755i)3-s + (0.723 + 0.690i)4-s + (−0.841 + 0.540i)5-s + (−0.888 + 0.458i)6-s + (0.786 − 0.618i)7-s + (−0.415 − 0.909i)8-s + (−0.142 − 0.989i)9-s + (0.981 − 0.189i)10-s + (0.888 + 0.458i)11-s + (0.995 − 0.0950i)12-s + (−0.580 − 0.814i)13-s + (−0.959 + 0.281i)14-s + (−0.142 + 0.989i)15-s + (0.0475 + 0.998i)16-s + (0.723 − 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5610476145 - 0.9295192667i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5610476145 - 0.9295192667i\) |
\(L(1)\) |
\(\approx\) |
\(0.7252745480 - 0.4265363599i\) |
\(L(1)\) |
\(\approx\) |
\(0.7252745480 - 0.4265363599i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 + (-0.928 - 0.371i)T \) |
| 3 | \( 1 + (0.654 - 0.755i)T \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 7 | \( 1 + (0.786 - 0.618i)T \) |
| 11 | \( 1 + (0.888 + 0.458i)T \) |
| 13 | \( 1 + (-0.580 - 0.814i)T \) |
| 17 | \( 1 + (0.723 - 0.690i)T \) |
| 19 | \( 1 + (-0.786 - 0.618i)T \) |
| 23 | \( 1 + (-0.327 - 0.945i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.580 + 0.814i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.235 - 0.971i)T \) |
| 43 | \( 1 + (0.959 + 0.281i)T \) |
| 47 | \( 1 + (0.981 + 0.189i)T \) |
| 53 | \( 1 + (0.959 - 0.281i)T \) |
| 59 | \( 1 + (0.415 + 0.909i)T \) |
| 61 | \( 1 + (0.888 - 0.458i)T \) |
| 71 | \( 1 + (0.723 + 0.690i)T \) |
| 73 | \( 1 + (-0.888 + 0.458i)T \) |
| 79 | \( 1 + (0.995 - 0.0950i)T \) |
| 83 | \( 1 + (0.0475 + 0.998i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−32.09328398339832467446057869264, −31.32957933735483450815838509734, −29.860880831267664776334348550081, −28.13492721223593510909139763098, −27.61905370572441595477739163150, −26.86663659965503113378868262921, −25.61163603215774297066664546217, −24.60184164568805518199810777109, −23.70636438738249480024658948276, −21.755146182575494648323140232048, −20.727301896500405852592696732757, −19.5629513661362795692961856673, −18.9132597949400911841120788251, −17.06493000648547477911897017220, −16.303027680290164817918205414017, −15.04408819537026068290329289648, −14.415280368711310766585683819050, −12.01678942869342707480120132312, −10.95530991337511658445257893609, −9.3589570583898969095614957333, −8.57423313786400086140873645892, −7.58426412831882347840407831489, −5.50871039635781330554736678444, −3.93417930761515235183729574590, −1.79908957334904189853951495431,
0.72529800243408573350276801595, 2.43375298507177956356641652321, 3.904558514586871407406272960810, 6.903932221329547815503413093694, 7.60975833590611258321863533021, 8.67201654228346442835457226020, 10.2546630712627529311113033600, 11.58224878675793990838914846151, 12.507714542568628918817249857062, 14.30093181465422821837577692349, 15.28008255159135450114175215603, 17.05111999769145553922965143039, 18.0122215349115256593991455977, 19.10494782775015569245598423704, 19.95358649157850374015011741883, 20.72568104318138695942088963572, 22.528594497904429513827203787001, 23.93120609561471523865612030685, 24.95506071733374285842586851771, 26.04452518065133350680738279005, 27.09840200499377036593957008748, 27.762406096917210986847244225304, 29.5283643599863572308340202933, 30.291325356695603839842265219143, 30.78823535000466228957224868520