L(s) = 1 | + (−0.0835 − 1.73i)3-s + (0.431 − 0.431i)5-s + 3.10·7-s + (−2.98 + 0.289i)9-s + (−2.98 − 2.98i)11-s + (2.10 − 2.10i)13-s + (−0.782 − 0.710i)15-s − 2.42i·17-s + (0.710 + 0.710i)19-s + (−0.259 − 5.36i)21-s + 5.97i·23-s + 4.62i·25-s + (0.749 + 5.14i)27-s + (2.86 + 2.86i)29-s + 0.524i·31-s + ⋯ |
L(s) = 1 | + (−0.0482 − 0.998i)3-s + (0.193 − 0.193i)5-s + 1.17·7-s + (−0.995 + 0.0963i)9-s + (−0.900 − 0.900i)11-s + (0.583 − 0.583i)13-s + (−0.202 − 0.183i)15-s − 0.589i·17-s + (0.163 + 0.163i)19-s + (−0.0565 − 1.17i)21-s + 1.24i·23-s + 0.925i·25-s + (0.144 + 0.989i)27-s + (0.531 + 0.531i)29-s + 0.0941i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.01392 - 0.717929i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.01392 - 0.717929i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0835 + 1.73i)T \) |
good | 5 | \( 1 + (-0.431 + 0.431i)T - 5iT^{2} \) |
| 7 | \( 1 - 3.10T + 7T^{2} \) |
| 11 | \( 1 + (2.98 + 2.98i)T + 11iT^{2} \) |
| 13 | \( 1 + (-2.10 + 2.10i)T - 13iT^{2} \) |
| 17 | \( 1 + 2.42iT - 17T^{2} \) |
| 19 | \( 1 + (-0.710 - 0.710i)T + 19iT^{2} \) |
| 23 | \( 1 - 5.97iT - 23T^{2} \) |
| 29 | \( 1 + (-2.86 - 2.86i)T + 29iT^{2} \) |
| 31 | \( 1 - 0.524iT - 31T^{2} \) |
| 37 | \( 1 + (-1.52 - 1.52i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.81T + 41T^{2} \) |
| 43 | \( 1 + (0.710 - 0.710i)T - 43iT^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 + (8.83 - 8.83i)T - 53iT^{2} \) |
| 59 | \( 1 + (-0.0804 - 0.0804i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.72 - 5.72i)T - 61iT^{2} \) |
| 67 | \( 1 + (-0.391 - 0.391i)T + 67iT^{2} \) |
| 71 | \( 1 + 5.01iT - 71T^{2} \) |
| 73 | \( 1 + 13.4iT - 73T^{2} \) |
| 79 | \( 1 - 3.47iT - 79T^{2} \) |
| 83 | \( 1 + (4.55 - 4.55i)T - 83iT^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 8.67T + 97T^{2} \) |
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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
−12.37559605001368659642250626966, −11.34899009804716546349452813548, −10.75843632344850269383223887473, −9.081234542439199021348014542921, −8.087702965265302598050647810682, −7.47230702808958582172601267170, −5.90325635891205152239981686176, −5.13281070745125878993116384227, −3.03620387649918954668145143874, −1.33883260711732846891386188754,
2.36168515248227211072694865671, 4.21317630814752928008318784895, 5.00472356211450086754314098747, 6.32144114407368144650365969126, 7.898598095107778997253900930029, 8.721922617620050675074717185623, 9.977381167347363572666738588398, 10.68910775581731328984195572037, 11.50707724274834309065155293552, 12.62608585153439345774444810368