L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s − 4-s − i·5-s + (−0.707 + 0.707i)6-s + i·8-s + i·9-s − 10-s + (−0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + 16-s + (−0.707 + 0.707i)17-s + 18-s + (−0.707 + 0.707i)19-s + ⋯ |
L(s) = 1 | − i·2-s + (−0.707 − 0.707i)3-s − 4-s − i·5-s + (−0.707 + 0.707i)6-s + i·8-s + i·9-s − 10-s + (−0.707 − 0.707i)11-s + (0.707 + 0.707i)12-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + 16-s + (−0.707 + 0.707i)17-s + 18-s + (−0.707 + 0.707i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1345525219 - 0.09184424007i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1345525219 - 0.09184424007i\) |
\(L(1)\) |
\(\approx\) |
\(0.3301980196 - 0.4283484947i\) |
\(L(1)\) |
\(\approx\) |
\(0.3301980196 - 0.4283484947i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 - iT \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 - iT \) |
| 11 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + (-0.707 - 0.707i)T \) |
| 17 | \( 1 + (-0.707 + 0.707i)T \) |
| 19 | \( 1 + (-0.707 + 0.707i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (0.707 + 0.707i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + iT \) |
| 67 | \( 1 + (0.707 - 0.707i)T \) |
| 71 | \( 1 + (0.707 + 0.707i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.707 - 0.707i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.707 - 0.707i)T \) |
| 97 | \( 1 + (0.707 - 0.707i)T \) |
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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
−26.55700913016902753409222704318, −25.55589389725908469039792265327, −24.34220604021405089052328505028, −23.362503610505379775695045947, −22.84623901451677877566594965045, −21.921067572341521450089356235760, −21.3645474169172130861839568601, −19.79652864715833076257056715699, −18.55233733679874674618397568773, −17.81564275888201962599126654955, −17.16267696782819268661822840233, −15.96332386118053182068575855605, −15.38284485013459916879572575916, −14.588918273027568086704039472194, −13.58104749637728482890562103002, −12.259078741108148294479318181678, −11.161266574345947997665721551466, −10.03601514387850589910498108767, −9.472045116682168879383247447793, −7.95941678208819945878566462860, −6.82690151850865803310258867206, −6.24853893922954146864299664474, −4.867361460076184781131147760046, −4.243880328251595586029124218756, −2.62839574552881041390116045274,
0.12055575090607238709663671511, 1.443713379873153790707833172771, 2.594231876108195957292395572987, 4.25454303983743704650102297979, 5.21950326286866553441495919440, 6.10644340925641820257543513222, 7.97510714710621169475163076319, 8.445332346558418745166066721150, 9.97215592561463513484518406326, 10.76937686432656532336167900483, 11.84464536315347208767589670660, 12.6555902107032729991879369692, 13.12693426989001374801650912867, 14.17655996223859897678321753652, 15.79301041928123578949344993143, 16.92243462052052454991896817737, 17.568763416641208943888685809098, 18.482663781884749590130211735232, 19.467263577968518505119990946008, 20.09271822778957199081843998187, 21.270128511859463002907371159607, 21.94388795840939007354977640640, 22.99962254805977754063083318135, 23.84896422782605492750425089872, 24.43663366168104874665418952488