L(s) = 1 | + (−0.320 + 0.947i)2-s + (−0.794 − 0.607i)4-s + (−0.818 − 0.574i)5-s + (−0.591 + 0.806i)7-s + (0.830 − 0.557i)8-s + (0.806 − 0.591i)10-s + (0.852 − 0.523i)11-s + (−0.917 + 0.396i)13-s + (−0.574 − 0.818i)14-s + (0.262 + 0.965i)16-s + (−0.909 + 0.415i)17-s + (0.607 − 0.794i)19-s + (0.301 + 0.953i)20-s + (0.222 + 0.974i)22-s + (0.339 + 0.940i)25-s + (−0.0815 − 0.996i)26-s + ⋯ |
L(s) = 1 | + (−0.320 + 0.947i)2-s + (−0.794 − 0.607i)4-s + (−0.818 − 0.574i)5-s + (−0.591 + 0.806i)7-s + (0.830 − 0.557i)8-s + (0.806 − 0.591i)10-s + (0.852 − 0.523i)11-s + (−0.917 + 0.396i)13-s + (−0.574 − 0.818i)14-s + (0.262 + 0.965i)16-s + (−0.909 + 0.415i)17-s + (0.607 − 0.794i)19-s + (0.301 + 0.953i)20-s + (0.222 + 0.974i)22-s + (0.339 + 0.940i)25-s + (−0.0815 − 0.996i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4366089060 - 0.1612077392i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4366089060 - 0.1612077392i\) |
\(L(1)\) |
\(\approx\) |
\(0.5767177781 + 0.1899847436i\) |
\(L(1)\) |
\(\approx\) |
\(0.5767177781 + 0.1899847436i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.320 + 0.947i)T \) |
| 5 | \( 1 + (-0.818 - 0.574i)T \) |
| 7 | \( 1 + (-0.591 + 0.806i)T \) |
| 11 | \( 1 + (0.852 - 0.523i)T \) |
| 13 | \( 1 + (-0.917 + 0.396i)T \) |
| 17 | \( 1 + (-0.909 + 0.415i)T \) |
| 19 | \( 1 + (0.607 - 0.794i)T \) |
| 31 | \( 1 + (-0.728 - 0.685i)T \) |
| 37 | \( 1 + (0.162 + 0.986i)T \) |
| 41 | \( 1 + (0.281 + 0.959i)T \) |
| 43 | \( 1 + (0.728 - 0.685i)T \) |
| 47 | \( 1 + (-0.433 + 0.900i)T \) |
| 53 | \( 1 + (-0.182 + 0.983i)T \) |
| 59 | \( 1 + (0.654 - 0.755i)T \) |
| 61 | \( 1 + (0.359 + 0.933i)T \) |
| 67 | \( 1 + (0.523 - 0.852i)T \) |
| 71 | \( 1 + (-0.992 + 0.122i)T \) |
| 73 | \( 1 + (0.891 - 0.452i)T \) |
| 79 | \( 1 + (-0.965 - 0.262i)T \) |
| 83 | \( 1 + (-0.488 + 0.872i)T \) |
| 89 | \( 1 + (-0.505 - 0.862i)T \) |
| 97 | \( 1 + (0.998 - 0.0611i)T \) |
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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
−20.006373060270480487204356209363, −19.48538871106568822715438663127, −18.80614833447931304215743524780, −17.855987779287423705928580382285, −17.40748610131107131946672860701, −16.43108155028539651392892818868, −15.86249331409675151630685977865, −14.56862362955997584136818954916, −14.29950416266944230608615281139, −13.19718831041213426689966595317, −12.48931250078271750586353670663, −11.87019845859445740867408815605, −11.12112478983466554387224162300, −10.41003513490062629702698440858, −9.7525078152659948370370398142, −9.05347393462561109665757459747, −7.990870367154042473027306339245, −7.24636431445663796594776639905, −6.79682007562852696559571960418, −5.31284192702015500545972647362, −4.236086743183731710789926052050, −3.806484694373605634293316733526, −2.96880462686636662342589162916, −2.06157250488764351220179445069, −0.82730034246669516263761970785,
0.25148478351596567882187052898, 1.4583081984329662701877843818, 2.801183867720110180246013924451, 3.94641690008916117694383859711, 4.61845508930082358832753393184, 5.46488997309926105635005393971, 6.33302590704143282862941175915, 7.005560978053061126401448299947, 7.83727348389031431968551920039, 8.71101353696041313838427786074, 9.18106956166597162076254802168, 9.75577852138853442335079217357, 11.09146474325765263143362557080, 11.73053057224306251892179548294, 12.63768766297350642694220025703, 13.25795521978416099931000523030, 14.219706048526428986489563205454, 15.03229794791440972313826436771, 15.528582905764067812523331349077, 16.23587498751661308217209248776, 16.82957294924703247523262442707, 17.48833768413442725888699747397, 18.44289846007128070459387500021, 19.19207647399298862369114993167, 19.59278278922818682644112059864