L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)6-s − 7-s − 8-s + (−0.5 + 0.866i)9-s + 11-s − 12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s − 18-s + (−0.5 − 0.866i)21-s + (0.5 + 0.866i)22-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)6-s − 7-s − 8-s + (−0.5 + 0.866i)9-s + 11-s − 12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s − 18-s + (−0.5 − 0.866i)21-s + (0.5 + 0.866i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5228970179 + 1.179861651i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5228970179 + 1.179861651i\) |
\(L(1)\) |
\(\approx\) |
\(0.9177439011 + 0.9417770619i\) |
\(L(1)\) |
\(\approx\) |
\(0.9177439011 + 0.9417770619i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 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
−29.8133426825595782339532768477, −29.141104109322435021123000749499, −28.10455416886462306958010631294, −26.72042037582687829703450264991, −25.49334583912964409728507953249, −24.4964474333522027818424570221, −23.30171340078827647152503981196, −22.58733812510470909383998418320, −21.23249625982532991277639728290, −20.15019445008209247633985769022, −19.197088708469135759285614073191, −18.68309814610733979929367930204, −17.16341706343034806031535889036, −15.4767701427701453221383025711, −14.08137989536986348306958312075, −13.49982687410953130145814621061, −12.27612072139091115288707950588, −11.50461000417158423687801756630, −9.69348740708964302684551174605, −8.92231891657506949402537761569, −6.989654342882925759241790926646, −5.94135634519053757600862516734, −3.944669071698011407123983561689, −2.83952056973052612580228467360, −1.296923330494725049029294919025,
3.1328440918878846592268258955, 3.98263028195598902515093218485, 5.49618933285189871085703651262, 6.69768439592008731741439859645, 8.28009902937974379752075157865, 9.21384484575818812090879747829, 10.491856211829518165136093476080, 12.31284408276088294332005466707, 13.4389128734075009573412254801, 14.564623696431030316508439949278, 15.43442627450889253563110671995, 16.40785034518957818681618069046, 17.240000964013814086249883244570, 18.95466079263623328417499419820, 20.168312657579990913112481408143, 21.333367182404571580267970770825, 22.400376307061993162315468901724, 22.92620465248974627870070736218, 24.55206526466649149219926218420, 25.50826530058006954680523496480, 26.115602647358281627958642063940, 27.23947900968980724808772030380, 28.15821971373866719577151029183, 29.88825001605465705447602872552, 30.84491014627610342653236084454