L(s) = 1 | + (0.803 − 1.16i)2-s + (−0.709 − 1.87i)4-s + (−1.29 − 1.82i)5-s + (−0.306 − 0.306i)7-s + (−2.74 − 0.676i)8-s + (−3.16 + 0.0386i)10-s − 4.06·11-s + (−0.625 − 0.625i)13-s + (−0.603 + 0.110i)14-s + (−2.99 + 2.65i)16-s + (3.57 − 3.57i)17-s + 6.82·19-s + (−2.49 + 3.71i)20-s + (−3.26 + 4.73i)22-s + (−1.58 − 1.58i)23-s + ⋯ |
L(s) = 1 | + (0.568 − 0.822i)2-s + (−0.354 − 0.935i)4-s + (−0.578 − 0.815i)5-s + (−0.115 − 0.115i)7-s + (−0.970 − 0.239i)8-s + (−0.999 + 0.0122i)10-s − 1.22·11-s + (−0.173 − 0.173i)13-s + (−0.161 + 0.0295i)14-s + (−0.748 + 0.663i)16-s + (0.867 − 0.867i)17-s + 1.56·19-s + (−0.557 + 0.829i)20-s + (−0.696 + 1.00i)22-s + (−0.331 − 0.331i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.192986 - 1.22363i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.192986 - 1.22363i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.803 + 1.16i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.29 + 1.82i)T \) |
good | 7 | \( 1 + (0.306 + 0.306i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 + (0.625 + 0.625i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.57 + 3.57i)T - 17iT^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 + (1.58 + 1.58i)T + 23iT^{2} \) |
| 29 | \( 1 + 8.50iT - 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 + (-1.69 + 1.69i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + (-4.18 - 4.18i)T + 43iT^{2} \) |
| 47 | \( 1 + (-4.58 + 4.58i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.41 + 7.41i)T - 53iT^{2} \) |
| 59 | \( 1 - 8.79iT - 59T^{2} \) |
| 61 | \( 1 + 6.08iT - 61T^{2} \) |
| 67 | \( 1 + (-6.18 + 6.18i)T - 67iT^{2} \) |
| 71 | \( 1 - 14.7iT - 71T^{2} \) |
| 73 | \( 1 + (-3.05 + 3.05i)T - 73iT^{2} \) |
| 79 | \( 1 - 8.56iT - 79T^{2} \) |
| 83 | \( 1 + (5.13 - 5.13i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 + (-1.75 - 1.75i)T + 97iT^{2} \) |
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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
−11.33695318163309888967184383695, −10.07065510647387222628301508809, −9.547716280270134744810750297876, −8.248315305385209104198791901367, −7.36818693820366461822856686841, −5.62660817800711434580895621401, −5.03488196613716218457251423834, −3.83171278154373471229678267198, −2.63428917739354465201455550310, −0.71505333258115221520062156171,
2.85575679476287398707241340903, 3.75391543177769836466347612001, 5.16891086838727256919247442404, 5.99866342754790174230446135583, 7.38129251496143432041085821374, 7.61490446737767256711222874929, 8.822518115478470729227254295176, 10.07199112975051585820975659785, 11.01916278005294379073359651612, 12.09791647619775210585660746178