L(s) = 1 | + (0.866 + 1.5i)5-s + (0.5 − 0.866i)7-s + (0.866 + 0.5i)11-s + (−1 + 1.73i)25-s − i·29-s + (−1.5 − 0.866i)31-s + 1.73·35-s + (−0.499 − 0.866i)49-s + (0.866 + 0.5i)53-s + 1.73i·55-s + (−0.866 + 1.5i)59-s + (0.866 − 0.499i)77-s + (0.5 + 0.866i)79-s − 1.73·83-s − 1.73i·97-s + ⋯ |
L(s) = 1 | + (0.866 + 1.5i)5-s + (0.5 − 0.866i)7-s + (0.866 + 0.5i)11-s + (−1 + 1.73i)25-s − i·29-s + (−1.5 − 0.866i)31-s + 1.73·35-s + (−0.499 − 0.866i)49-s + (0.866 + 0.5i)53-s + 1.73i·55-s + (−0.866 + 1.5i)59-s + (0.866 − 0.499i)77-s + (0.5 + 0.866i)79-s − 1.73·83-s − 1.73i·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.474528847\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.474528847\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + 1.73iT - T^{2} \) |
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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
−9.720107363261306515883837085759, −8.721864357359744306425747031840, −7.45292848686517787503435491901, −7.21644970861672826768641830720, −6.31254023219666110765394716893, −5.66040613165431034923401088686, −4.37773044823573934479972895224, −3.63838132946721242100265009044, −2.50744296080747164808441305408, −1.59517689284999459059657863235,
1.28797238410692487110327069689, 2.04175386152273353527504027886, 3.45555184705016807768510337057, 4.62312554575023135357416503864, 5.30257849445462530020878861480, 5.84325780958923743493269338971, 6.77017211782634985405107076379, 7.970918842062309682411828197433, 8.837947718689458514013551694901, 8.968449389284427238641763482313