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
−24.43663366168104874665418952488, −23.84896422782605492750425089872, −22.99962254805977754063083318135, −21.94388795840939007354977640640, −21.270128511859463002907371159607, −20.09271822778957199081843998187, −19.467263577968518505119990946008, −18.482663781884749590130211735232, −17.568763416641208943888685809098, −16.92243462052052454991896817737, −15.79301041928123578949344993143, −14.17655996223859897678321753652, −13.12693426989001374801650912867, −12.6555902107032729991879369692, −11.84464536315347208767589670660, −10.76937686432656532336167900483, −9.97215592561463513484518406326, −8.445332346558418745166066721150, −7.97510714710621169475163076319, −6.10644340925641820257543513222, −5.21950326286866553441495919440, −4.25454303983743704650102297979, −2.594231876108195957292395572987, −1.443713379873153790707833172771, −0.12055575090607238709663671511,
2.62839574552881041390116045274, 4.243880328251595586029124218756, 4.867361460076184781131147760046, 6.24853893922954146864299664474, 6.82690151850865803310258867206, 7.95941678208819945878566462860, 9.472045116682168879383247447793, 10.03601514387850589910498108767, 11.161266574345947997665721551466, 12.259078741108148294479318181678, 13.58104749637728482890562103002, 14.588918273027568086704039472194, 15.38284485013459916879572575916, 15.96332386118053182068575855605, 17.16267696782819268661822840233, 17.81564275888201962599126654955, 18.55233733679874674618397568773, 19.79652864715833076257056715699, 21.3645474169172130861839568601, 21.921067572341521450089356235760, 22.84623901451677877566594965045, 23.362503610505379775695045947, 24.34220604021405089052328505028, 25.55589389725908469039792265327, 26.55700913016902753409222704318