L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.286 − 0.957i)3-s + (−0.5 − 0.866i)4-s + (−0.893 + 0.448i)5-s + (−0.973 − 0.230i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (−0.686 + 0.727i)12-s + (−0.893 + 0.448i)13-s + (−0.893 + 0.448i)14-s + (0.686 + 0.727i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.286 − 0.957i)3-s + (−0.5 − 0.866i)4-s + (−0.893 + 0.448i)5-s + (−0.973 − 0.230i)6-s + (−0.835 − 0.549i)7-s − 8-s + (−0.835 + 0.549i)9-s + (−0.0581 + 0.998i)10-s + (−0.0581 − 0.998i)11-s + (−0.686 + 0.727i)12-s + (−0.893 + 0.448i)13-s + (−0.893 + 0.448i)14-s + (0.686 + 0.727i)15-s + (−0.5 + 0.866i)16-s − 17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2596145076 - 0.7565928077i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2596145076 - 0.7565928077i\) |
\(L(1)\) |
\(\approx\) |
\(0.5227920538 - 0.5491195402i\) |
\(L(1)\) |
\(\approx\) |
\(0.5227920538 - 0.5491195402i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.286 - 0.957i)T \) |
| 5 | \( 1 + (-0.893 + 0.448i)T \) |
| 7 | \( 1 + (-0.835 - 0.549i)T \) |
| 11 | \( 1 + (-0.0581 - 0.998i)T \) |
| 13 | \( 1 + (-0.893 + 0.448i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (0.939 + 0.342i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.893 + 0.448i)T \) |
| 31 | \( 1 + (0.993 + 0.116i)T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (-0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.973 - 0.230i)T \) |
| 53 | \( 1 + (-0.286 - 0.957i)T \) |
| 59 | \( 1 + (0.686 + 0.727i)T \) |
| 61 | \( 1 + (-0.396 + 0.918i)T \) |
| 67 | \( 1 + (-0.686 - 0.727i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (0.893 + 0.448i)T \) |
| 79 | \( 1 + (-0.973 + 0.230i)T \) |
| 83 | \( 1 + (0.597 + 0.802i)T \) |
| 89 | \( 1 + (-0.893 + 0.448i)T \) |
| 97 | \( 1 + (0.993 - 0.116i)T \) |
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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
−18.65145410790373161192039025661, −17.649459347076660887319551735654, −17.21123039820074194107115016999, −16.51691461516948972306636377986, −15.77957735600928759240109834513, −15.40781671511027976840036073484, −15.084695812504640483214251887818, −14.24435718053472560437689777208, −13.1037146602591088619213568187, −12.68469713995672837023179142379, −11.93947099629482302851314710359, −11.46042028716866121296579764733, −10.35289655581314666361345326261, −9.34123530753672905048250626672, −9.23287215214680198440836874924, −8.26830759589151343015944920683, −7.40875452161196273079194138477, −6.833852951248748079925637026547, −5.90778289620917610885765497384, −5.12975945971158931933238070731, −4.642477440616996709699483440312, −4.03365114260066436272065841411, −3.1115465352796463459859703265, −2.55717006858597008695986615929, −0.482905594319408439173582052400,
0.441135636302694677296015123221, 1.235431925557045877357411739499, 2.43017125587404437992257536820, 3.029877284015256464100592105613, 3.65981729832505399419886259156, 4.57112933663438747021496514265, 5.425554777495964657679578540116, 6.272407856366921412537283385830, 6.934062699787029497048819326449, 7.48070334044632824866875664130, 8.50799019028504367710451822009, 9.23491370458579321747764299666, 10.16410014454425408414054520805, 10.95616404780153615765048208261, 11.378169811017733117294212855049, 12.029026947367532561880855305232, 12.63127032701645472540433242651, 13.46657999634845083756952178836, 13.74393609892962639512012132745, 14.59209587082857439127793648752, 15.32641252050823207783325944889, 16.23338345348036241353766446483, 16.83487739380637542144356309186, 17.772949177221726775682937792175, 18.53133382089129339432810999566