L(s) = 1 | + (0.932 − 0.361i)2-s + (0.445 − 0.895i)3-s + (0.739 − 0.673i)4-s + (0.445 − 0.895i)5-s + (0.0922 − 0.995i)6-s + (0.0922 − 0.995i)7-s + (0.445 − 0.895i)8-s + (−0.602 − 0.798i)9-s + (0.0922 − 0.995i)10-s + (−0.602 + 0.798i)11-s + (−0.273 − 0.961i)12-s + (0.739 + 0.673i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 − 0.995i)16-s + (−0.850 − 0.526i)17-s + ⋯ |
L(s) = 1 | + (0.932 − 0.361i)2-s + (0.445 − 0.895i)3-s + (0.739 − 0.673i)4-s + (0.445 − 0.895i)5-s + (0.0922 − 0.995i)6-s + (0.0922 − 0.995i)7-s + (0.445 − 0.895i)8-s + (−0.602 − 0.798i)9-s + (0.0922 − 0.995i)10-s + (−0.602 + 0.798i)11-s + (−0.273 − 0.961i)12-s + (0.739 + 0.673i)13-s + (−0.273 − 0.961i)14-s + (−0.602 − 0.798i)15-s + (0.0922 − 0.995i)16-s + (−0.850 − 0.526i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7964507528 - 3.164779353i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7964507528 - 3.164779353i\) |
\(L(1)\) |
\(\approx\) |
\(1.501859822 - 1.581709635i\) |
\(L(1)\) |
\(\approx\) |
\(1.501859822 - 1.581709635i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 919 | \( 1 \) |
good | 2 | \( 1 + (0.932 - 0.361i)T \) |
| 3 | \( 1 + (0.445 - 0.895i)T \) |
| 5 | \( 1 + (0.445 - 0.895i)T \) |
| 7 | \( 1 + (0.0922 - 0.995i)T \) |
| 11 | \( 1 + (-0.602 + 0.798i)T \) |
| 13 | \( 1 + (0.739 + 0.673i)T \) |
| 17 | \( 1 + (-0.850 - 0.526i)T \) |
| 19 | \( 1 + (0.445 + 0.895i)T \) |
| 23 | \( 1 + (-0.602 + 0.798i)T \) |
| 29 | \( 1 + (0.932 - 0.361i)T \) |
| 31 | \( 1 + (0.0922 - 0.995i)T \) |
| 37 | \( 1 + (0.739 + 0.673i)T \) |
| 41 | \( 1 + (-0.273 + 0.961i)T \) |
| 43 | \( 1 + (0.445 + 0.895i)T \) |
| 47 | \( 1 + (0.739 - 0.673i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.739 - 0.673i)T \) |
| 61 | \( 1 + (0.0922 + 0.995i)T \) |
| 67 | \( 1 + (-0.982 + 0.183i)T \) |
| 71 | \( 1 + (0.0922 - 0.995i)T \) |
| 73 | \( 1 + (-0.982 + 0.183i)T \) |
| 79 | \( 1 + (0.0922 + 0.995i)T \) |
| 83 | \( 1 + (-0.850 + 0.526i)T \) |
| 89 | \( 1 + (0.445 + 0.895i)T \) |
| 97 | \( 1 + (-0.602 - 0.798i)T \) |
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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
−22.01202824447161632698773143213, −21.699938955709610777435808417657, −20.98355640881974205472366688190, −20.125079885419642376101228813778, −19.19843273970880162361143094927, −18.11826508090544861295066603734, −17.46683319273325870974631632797, −16.08494353739971662581337163550, −15.73526107277085275675356833026, −15.09959338829302999024780666441, −14.25982169931639397041915658483, −13.67099796131778183175046388610, −12.84613269411904152152791243034, −11.645709440488343289072023185629, −10.80178220590229133426177554419, −10.39826100161340565898089443053, −8.84320718055087149765568919586, −8.45218052823395539656972089854, −7.25917797762092572822814650759, −6.06112180119094633515608264914, −5.64774587852749451792857703771, −4.65537547292291325800690934624, −3.52792065478387674011497707456, −2.79919797300751038118393939580, −2.23907177604948498452914362966,
0.9872706712559047676299441622, 1.72781713215639191602516957977, 2.61909846552189969848072237277, 3.93081957823840250962201186984, 4.55724625359926148821018575020, 5.7160596014020970885426989469, 6.51206328984039719417954822227, 7.41723999251925249182302512147, 8.21113178498467135423043836361, 9.490215507179628008332361352479, 10.11746443907118858309146892773, 11.41853786249265948593542078185, 12.00285063570402993708783546280, 13.006686791606176056650130958130, 13.506250873776021417215296101291, 13.883300896368459486480582188222, 14.90444040768533222774983412348, 15.9292450401692594505113957708, 16.68997732087690915276283324741, 17.757354885753064869760249481532, 18.41437194870837527186842199463, 19.58856851105002016137990647846, 20.143644618229698994255352799127, 20.70507402915647372324562837695, 21.231700729636116378643340851900