L(s) = 1 | + 2.70i·3-s + (−1.60 − 1.60i)7-s − 4.33·9-s + (3.90 + 3.90i)11-s + 1.48·13-s + (4.26 + 4.26i)17-s + (0.162 + 0.162i)19-s + (4.35 − 4.35i)21-s + (−5.87 + 5.87i)23-s − 3.60i·27-s + (1.48 − 1.48i)29-s − 6.78i·31-s + (−10.5 + 10.5i)33-s − 6.13·37-s + 4.02i·39-s + ⋯ |
L(s) = 1 | + 1.56i·3-s + (−0.608 − 0.608i)7-s − 1.44·9-s + (1.17 + 1.17i)11-s + 0.412·13-s + (1.03 + 1.03i)17-s + (0.0373 + 0.0373i)19-s + (0.950 − 0.950i)21-s + (−1.22 + 1.22i)23-s − 0.694i·27-s + (0.276 − 0.276i)29-s − 1.21i·31-s + (−1.84 + 1.84i)33-s − 1.00·37-s + 0.644i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.385665005\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.385665005\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 2.70iT - 3T^{2} \) |
| 7 | \( 1 + (1.60 + 1.60i)T + 7iT^{2} \) |
| 11 | \( 1 + (-3.90 - 3.90i)T + 11iT^{2} \) |
| 13 | \( 1 - 1.48T + 13T^{2} \) |
| 17 | \( 1 + (-4.26 - 4.26i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.162 - 0.162i)T + 19iT^{2} \) |
| 23 | \( 1 + (5.87 - 5.87i)T - 23iT^{2} \) |
| 29 | \( 1 + (-1.48 + 1.48i)T - 29iT^{2} \) |
| 31 | \( 1 + 6.78iT - 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 - 2.75iT - 41T^{2} \) |
| 43 | \( 1 - 3.39T + 43T^{2} \) |
| 47 | \( 1 + (9.44 - 9.44i)T - 47iT^{2} \) |
| 53 | \( 1 - 1.54iT - 53T^{2} \) |
| 59 | \( 1 + (2.53 - 2.53i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.600 - 0.600i)T + 61iT^{2} \) |
| 67 | \( 1 + 8.14T + 67T^{2} \) |
| 71 | \( 1 - 4.55T + 71T^{2} \) |
| 73 | \( 1 + (-4.84 - 4.84i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.455T + 79T^{2} \) |
| 83 | \( 1 + 2.84iT - 83T^{2} \) |
| 89 | \( 1 + 1.91T + 89T^{2} \) |
| 97 | \( 1 + (1.73 + 1.73i)T + 97iT^{2} \) |
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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
−9.743000906987695598168838547109, −9.413673795364495450202389269911, −8.301520491136726232890197154713, −7.42353832297949596741875656556, −6.34959718749901503739030595727, −5.64386337682242873698932242089, −4.45466014076562661761401783788, −3.93230074291751513868898490297, −3.33161427496629447239037472530, −1.60465801435892763602165412452,
0.55669183016719040713138537849, 1.63536112308338226411831334641, 2.83901270002440539710120465117, 3.62431815774421450628765076866, 5.22642094369514100904494868979, 6.17814540693256460774773597755, 6.50942257138015494487867393549, 7.34100918590045976717743819491, 8.369938136094331871914558606965, 8.728471483245789525010797266284