L(s) = 1 | + (2.18 + 2.18i)3-s + (−2.18 − 0.456i)5-s + (1.79 − 1.79i)7-s + 6.58i·9-s + 0.913i·11-s + (−1.73 + 1.73i)13-s + (−3.79 − 5.79i)15-s + (3 + 3i)17-s − 3.46·19-s + 7.84·21-s + (3.79 + 3.79i)23-s + (4.58 + 1.99i)25-s + (−7.84 + 7.84i)27-s + 5.29i·29-s + 7.58i·31-s + ⋯ |
L(s) = 1 | + (1.26 + 1.26i)3-s + (−0.978 − 0.204i)5-s + (0.677 − 0.677i)7-s + 2.19i·9-s + 0.275i·11-s + (−0.480 + 0.480i)13-s + (−0.978 − 1.49i)15-s + (0.727 + 0.727i)17-s − 0.794·19-s + 1.71·21-s + (0.790 + 0.790i)23-s + (0.916 + 0.399i)25-s + (−1.50 + 1.50i)27-s + 0.982i·29-s + 1.36i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.106732765\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.106732765\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.18 + 0.456i)T \) |
good | 3 | \( 1 + (-2.18 - 2.18i)T + 3iT^{2} \) |
| 7 | \( 1 + (-1.79 + 1.79i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.913iT - 11T^{2} \) |
| 13 | \( 1 + (1.73 - 1.73i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3 - 3i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + (-3.79 - 3.79i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.29iT - 29T^{2} \) |
| 31 | \( 1 - 7.58iT - 31T^{2} \) |
| 37 | \( 1 + (5.19 + 5.19i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 + (0.361 + 0.361i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.79 + 3.79i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.64 - 2.64i)T - 53iT^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 - 6.20T + 61T^{2} \) |
| 67 | \( 1 + (-0.361 + 0.361i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.41iT - 71T^{2} \) |
| 73 | \( 1 + (-8.58 + 8.58i)T - 73iT^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + (-3.10 - 3.10i)T + 83iT^{2} \) |
| 89 | \( 1 + 15.1iT - 89T^{2} \) |
| 97 | \( 1 + (8.58 + 8.58i)T + 97iT^{2} \) |
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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
−9.916721787452758379609082461661, −8.859163308784289487345969646462, −8.563809766257111304848499751199, −7.61606026907495915503868146519, −7.08839808945313282936833197098, −5.21091817893335988232345857490, −4.58064357418883232923225038575, −3.82964992408798899629724745625, −3.18301493576272530702818587033, −1.71644503586669524456608904952,
0.77721436915355036961649964790, 2.27759991576514638845229845020, 2.89419499199611230187226299300, 3.96169076421326649443595374970, 5.18396030178573080996403696294, 6.43848757059042342371469713838, 7.20779144128685822563524358987, 7.976122988161769952035980937145, 8.307210221496525243143983670314, 9.042810440542378011357861053336