L(s) = 1 | + (0.213 − 0.976i)2-s + (−0.908 − 0.417i)4-s + (−0.0307 − 0.999i)5-s + (−0.389 + 0.920i)7-s + (−0.602 + 0.798i)8-s + (−0.982 − 0.183i)10-s + (0.213 − 0.976i)11-s + (0.992 − 0.122i)13-s + (0.816 + 0.577i)14-s + (0.650 + 0.759i)16-s + (0.445 − 0.895i)17-s + (0.932 − 0.361i)19-s + (−0.389 + 0.920i)20-s + (−0.908 − 0.417i)22-s + (0.213 + 0.976i)23-s + ⋯ |
L(s) = 1 | + (0.213 − 0.976i)2-s + (−0.908 − 0.417i)4-s + (−0.0307 − 0.999i)5-s + (−0.389 + 0.920i)7-s + (−0.602 + 0.798i)8-s + (−0.982 − 0.183i)10-s + (0.213 − 0.976i)11-s + (0.992 − 0.122i)13-s + (0.816 + 0.577i)14-s + (0.650 + 0.759i)16-s + (0.445 − 0.895i)17-s + (0.932 − 0.361i)19-s + (−0.389 + 0.920i)20-s + (−0.908 − 0.417i)22-s + (0.213 + 0.976i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4234338108 - 1.352969517i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4234338108 - 1.352969517i\) |
\(L(1)\) |
\(\approx\) |
\(0.8169299416 - 0.7335683292i\) |
\(L(1)\) |
\(\approx\) |
\(0.8169299416 - 0.7335683292i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 103 | \( 1 \) |
good | 2 | \( 1 + (0.213 - 0.976i)T \) |
| 5 | \( 1 + (-0.0307 - 0.999i)T \) |
| 7 | \( 1 + (-0.389 + 0.920i)T \) |
| 11 | \( 1 + (0.213 - 0.976i)T \) |
| 13 | \( 1 + (0.992 - 0.122i)T \) |
| 17 | \( 1 + (0.445 - 0.895i)T \) |
| 19 | \( 1 + (0.932 - 0.361i)T \) |
| 23 | \( 1 + (0.213 + 0.976i)T \) |
| 29 | \( 1 + (0.881 + 0.473i)T \) |
| 31 | \( 1 + (0.332 - 0.943i)T \) |
| 37 | \( 1 + (-0.273 + 0.961i)T \) |
| 41 | \( 1 + (0.881 - 0.473i)T \) |
| 43 | \( 1 + (-0.696 + 0.717i)T \) |
| 47 | \( 1 + (-0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.932 - 0.361i)T \) |
| 59 | \( 1 + (-0.389 - 0.920i)T \) |
| 61 | \( 1 + (-0.998 + 0.0615i)T \) |
| 67 | \( 1 + (-0.389 - 0.920i)T \) |
| 71 | \( 1 + (-0.850 - 0.526i)T \) |
| 73 | \( 1 + (-0.850 - 0.526i)T \) |
| 79 | \( 1 + (0.881 + 0.473i)T \) |
| 83 | \( 1 + (-0.389 + 0.920i)T \) |
| 89 | \( 1 + (0.0922 - 0.995i)T \) |
| 97 | \( 1 + (0.552 - 0.833i)T \) |
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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
−22.551293339276712359728034895474, −21.58569936921913278027806078114, −20.74419916501779550379774596593, −19.665603729804938705581260684680, −18.8717715923832113649744868385, −17.97584628767880106957978822695, −17.50631531565870592183296655259, −16.47505489723437324906078054863, −15.86657323554760322806693107148, −14.938831075200244678733936432289, −14.28199000643560380016416232185, −13.673393615745689422663000052312, −12.75348691337745400479032217641, −11.8712253430974950831725090538, −10.518345666635041111962766095876, −10.11760105259966471360367788510, −9.0120687446702130436945685157, −7.93025105916832783518119365857, −7.22746474410085622022573310578, −6.5351693946348782008216943317, −5.85767554220463844134166886378, −4.48350357337956956135486473131, −3.80554731469418122986926477503, −2.93899631814504057137137849616, −1.2331007472915699555276719194,
0.69362333657227313612166411058, 1.578736014706533276784130563563, 2.95479022388864998180681933, 3.50999980114408082863133166594, 4.79843347332299745708974750663, 5.48872814663653748831686477142, 6.21449057995311524986131521521, 7.91805231542792660664881704405, 8.7842757972828938271582942571, 9.25543382967509492191731369343, 10.092043294984864751556057650623, 11.50090463570436971103414585183, 11.649596669973752572143702652102, 12.63999009202635877138121335555, 13.506517730750658998720357717283, 13.85658171998821375465690812294, 15.245085321385004481182173852227, 15.97548698523185499530197108182, 16.74142871974944137953201708480, 17.93288944489133650653414264568, 18.47978383733531828219411394419, 19.36244463712390879156994612402, 19.95924626208793207898779100950, 20.93016084875219287916001431197, 21.321224235692451833160640884793