L(s) = 1 | + (0.971 − 1.02i)2-s + (−0.114 − 1.99i)4-s + 2.21·5-s − i·7-s + (−2.16 − 1.82i)8-s + (2.14 − 2.27i)10-s − 2.42i·11-s − 1.11i·13-s + (−1.02 − 0.971i)14-s + (−3.97 + 0.456i)16-s − 0.701i·17-s + 0.938·19-s + (−0.252 − 4.41i)20-s + (−2.49 − 2.35i)22-s + 4.30·23-s + ⋯ |
L(s) = 1 | + (0.686 − 0.727i)2-s + (−0.0571 − 0.998i)4-s + 0.989·5-s − 0.377i·7-s + (−0.765 − 0.643i)8-s + (0.679 − 0.719i)10-s − 0.730i·11-s − 0.309i·13-s + (−0.274 − 0.259i)14-s + (−0.993 + 0.114i)16-s − 0.170i·17-s + 0.215·19-s + (−0.0565 − 0.988i)20-s + (−0.531 − 0.501i)22-s + 0.897·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.643 + 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.671954039\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.671954039\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.971 + 1.02i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 2.21T + 5T^{2} \) |
| 11 | \( 1 + 2.42iT - 11T^{2} \) |
| 13 | \( 1 + 1.11iT - 13T^{2} \) |
| 17 | \( 1 + 0.701iT - 17T^{2} \) |
| 19 | \( 1 - 0.938T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 + 7.02iT - 31T^{2} \) |
| 37 | \( 1 - 3.12iT - 37T^{2} \) |
| 41 | \( 1 + 0.157iT - 41T^{2} \) |
| 43 | \( 1 - 7.08T + 43T^{2} \) |
| 47 | \( 1 + 0.867T + 47T^{2} \) |
| 53 | \( 1 + 3.91T + 53T^{2} \) |
| 59 | \( 1 + 7.28iT - 59T^{2} \) |
| 61 | \( 1 - 4.57iT - 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 + 6.93T + 71T^{2} \) |
| 73 | \( 1 - 7.02T + 73T^{2} \) |
| 79 | \( 1 - 6.65iT - 79T^{2} \) |
| 83 | \( 1 + 8.41iT - 83T^{2} \) |
| 89 | \( 1 - 7.34iT - 89T^{2} \) |
| 97 | \( 1 - 7.99T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.579979246182600311879807282879, −8.620537726454117796873368307066, −7.45915189501813518969158245927, −6.41568699526754509939366878124, −5.75270276846335844726199080424, −5.06167656941737574841989359918, −3.96641637233095976302101685827, −3.03098474525759442385435000929, −2.05420690578821436280402188741, −0.836216005303567950198516573384,
1.83758478918113025153664333576, 2.83074583128789796761389420484, 3.98768052172124422233184832529, 5.02576699701692671864436967753, 5.59039725257036146454707592900, 6.47137436124757562469151878396, 7.13338734364673507796244633226, 8.012324435267751900681790479619, 9.070490306664945429728536688457, 9.426408480638858960385438204021