L(s) = 1 | + (−0.637 − 1.26i)2-s + 2.87i·3-s + (−1.18 + 1.61i)4-s + i·5-s + (3.62 − 1.83i)6-s + 2.68·7-s + (2.78 + 0.469i)8-s − 5.25·9-s + (1.26 − 0.637i)10-s + 4.89·11-s + (−4.62 − 3.40i)12-s + 0.727·13-s + (−1.71 − 3.38i)14-s − 2.87·15-s + (−1.18 − 3.82i)16-s + 6.34i·17-s + ⋯ |
L(s) = 1 | + (−0.451 − 0.892i)2-s + 1.65i·3-s + (−0.593 + 0.805i)4-s + 0.447i·5-s + (1.48 − 0.748i)6-s + 1.01·7-s + (0.986 + 0.166i)8-s − 1.75·9-s + (0.399 − 0.201i)10-s + 1.47·11-s + (−1.33 − 0.983i)12-s + 0.201·13-s + (−0.457 − 0.904i)14-s − 0.741·15-s + (−0.296 − 0.955i)16-s + 1.53i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.883714 + 0.755587i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.883714 + 0.755587i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.637 + 1.26i)T \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (2.20 + 4.25i)T \) |
good | 3 | \( 1 - 2.87iT - 3T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 0.727T + 13T^{2} \) |
| 17 | \( 1 - 6.34iT - 17T^{2} \) |
| 19 | \( 1 + 2.68T + 19T^{2} \) |
| 29 | \( 1 + 4.27T + 29T^{2} \) |
| 31 | \( 1 - 6.13iT - 31T^{2} \) |
| 37 | \( 1 - 3.87iT - 37T^{2} \) |
| 41 | \( 1 + 8.07T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 + 9.80iT - 47T^{2} \) |
| 53 | \( 1 + 11.1iT - 53T^{2} \) |
| 59 | \( 1 + 1.87iT - 59T^{2} \) |
| 61 | \( 1 + 6.93iT - 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 - 7.24iT - 71T^{2} \) |
| 73 | \( 1 - 7.90T + 73T^{2} \) |
| 79 | \( 1 - 3.67T + 79T^{2} \) |
| 83 | \( 1 + 4.75T + 83T^{2} \) |
| 89 | \( 1 - 8.12iT - 89T^{2} \) |
| 97 | \( 1 - 2.68iT - 97T^{2} \) |
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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
−11.02457798172585938738654350379, −10.44713590264272887948900423033, −9.705490566742153116154212738821, −8.690378559749502184710735522197, −8.289173837012681394972991846965, −6.57845048499322678505906159552, −5.10867706689975881461190826520, −4.06760509523906057879190511563, −3.61969467538430802182978071041, −1.88627315307182738977593327741,
0.935773567118324688943439924890, 1.92068470097365812661544365976, 4.29468447638864748226167205008, 5.56932661524021951031399582670, 6.37762701144675591055903747428, 7.37966631933141412783340738454, 7.80571584002084938316903376228, 8.850284617814144157429086359183, 9.436102967628805492585586704974, 11.15471462619288544311908299721