L(s) = 1 | + (0.541 + 1.66i)2-s + (−1.67 + 1.21i)4-s + (0.598 − 1.84i)5-s + (−1.51 − 1.10i)8-s + (0.309 + 0.951i)9-s + 3.39·10-s + (0.876 + 0.481i)11-s + (−0.263 − 0.809i)13-s + (0.377 − 1.16i)16-s + (−1.41 + 1.03i)18-s + (1.50 + 1.09i)19-s + (1.24 + 3.81i)20-s + (−0.328 + 1.72i)22-s − 1.85·23-s + (−2.22 − 1.61i)25-s + (1.20 − 0.877i)26-s + ⋯ |
L(s) = 1 | + (0.541 + 1.66i)2-s + (−1.67 + 1.21i)4-s + (0.598 − 1.84i)5-s + (−1.51 − 1.10i)8-s + (0.309 + 0.951i)9-s + 3.39·10-s + (0.876 + 0.481i)11-s + (−0.263 − 0.809i)13-s + (0.377 − 1.16i)16-s + (−1.41 + 1.03i)18-s + (1.50 + 1.09i)19-s + (1.24 + 3.81i)20-s + (−0.328 + 1.72i)22-s − 1.85·23-s + (−2.22 − 1.61i)25-s + (1.20 − 0.877i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0798 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0798 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.346680345\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.346680345\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.876 - 0.481i)T \) |
| 79 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (-0.541 - 1.66i)T + (-0.809 + 0.587i)T^{2} \) |
| 3 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.598 + 1.84i)T + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.263 + 0.809i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-1.50 - 1.09i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + 1.85T + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.115 + 0.356i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + 1.27T + T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.866 - 0.629i)T + (0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 1.98T + T^{2} \) |
| 97 | \( 1 + (-0.0388 - 0.119i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−10.02658589870736109183095816347, −9.571717396128133845660847300879, −8.488083153939632112703744712056, −8.008010041805699104317797654101, −7.26186887539767158518470965761, −5.84487862237283949650916682059, −5.56067774194717251475956500841, −4.66270863282930419527159743404, −3.97372770609376117759918627080, −1.66998238131836381551886017037,
1.60163181060002706234892674355, 2.73526222556662928273110341323, 3.44774920933943332765024053498, 4.24045470478494384940280440715, 5.73653217622183876146735468055, 6.52875503996804492650855853694, 7.34807217927306274058153958827, 9.178241516124247872750125965563, 9.595322356291221475251135583574, 10.28474129034656348575270592785