L(s) = 1 | + i·2-s + (−0.535 + 0.309i)5-s + (0.809 − 1.40i)7-s + i·8-s + (−0.309 − 0.535i)10-s − 0.618i·11-s + (−0.309 + 0.535i)13-s + (1.40 + 0.809i)14-s − 16-s + (0.535 + 0.309i)17-s + (0.809 + 1.40i)19-s + 0.618·22-s + (1.40 − 0.809i)23-s + (−0.309 + 0.535i)25-s + (−0.535 − 0.309i)26-s + ⋯ |
L(s) = 1 | + i·2-s + (−0.535 + 0.309i)5-s + (0.809 − 1.40i)7-s + i·8-s + (−0.309 − 0.535i)10-s − 0.618i·11-s + (−0.309 + 0.535i)13-s + (1.40 + 0.809i)14-s − 16-s + (0.535 + 0.309i)17-s + (0.809 + 1.40i)19-s + 0.618·22-s + (1.40 − 0.809i)23-s + (−0.309 + 0.535i)25-s + (−0.535 − 0.309i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.182099374\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.182099374\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 + (0.535 - 0.309i)T + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + 0.618iT - T^{2} \) |
| 13 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.535 - 0.309i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.40 + 0.809i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + 1.61iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (1.40 - 0.809i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 - 1.61iT - T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 0.618T + T^{2} \) |
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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
−10.40977166466537435689668295970, −9.048149551768900452207017197279, −8.161261810186811529538458274503, −7.43522423122520699791576835778, −7.18846730466454868907695738276, −6.07588252402593871540576716246, −5.17335765987884501895308113640, −4.19227203711983259711145917073, −3.21557679055777571319286402888, −1.50876603173756198050445163025,
1.34277626155514618092388836474, 2.55963581647521991610905846354, 3.23053993401973310180864534127, 4.73247797042089308650219592689, 5.18882804741375075011808108133, 6.51669372257204513858150578447, 7.52693663674335699227913188535, 8.231732030862949509584898501271, 9.357162233163793996547777323172, 9.664888512521580693422632974020