L(s) = 1 | + (0.707 − 2.12i)5-s + (−2 − 2i)7-s + 2.82i·11-s + (3 − 3i)13-s + (−1.41 + 1.41i)17-s − 4i·19-s + (−5.65 − 5.65i)23-s + (−3.99 − 3i)25-s − 9.89·29-s + 8·31-s + (−5.65 + 2.82i)35-s + (−3 − 3i)37-s + 1.41i·41-s + (8.48 − 8.48i)47-s + i·49-s + ⋯ |
L(s) = 1 | + (0.316 − 0.948i)5-s + (−0.755 − 0.755i)7-s + 0.852i·11-s + (0.832 − 0.832i)13-s + (−0.342 + 0.342i)17-s − 0.917i·19-s + (−1.17 − 1.17i)23-s + (−0.799 − 0.600i)25-s − 1.83·29-s + 1.43·31-s + (−0.956 + 0.478i)35-s + (−0.493 − 0.493i)37-s + 0.220i·41-s + (1.23 − 1.23i)47-s + 0.142i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.660246 - 0.978521i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.660246 - 0.978521i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 2.12i)T \) |
good | 7 | \( 1 + (2 + 2i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.41 - 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (5.65 + 5.65i)T + 23iT^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (-8.48 + 8.48i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.07 - 7.07i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (8 + 8i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-7 + 7i)T - 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (-8.48 - 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 - 1.41T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−10.16974024552876129119546476722, −9.300519115297908930438498242872, −8.513647468645948387704149723777, −7.56753713705238076935838029161, −6.55562903589419399218114980432, −5.69862461239992987827074594328, −4.54940342328145707829952932644, −3.74495328496290865116212101767, −2.17944852982505820453673977806, −0.59829248653955591051148185110,
1.91820950984907582639690913729, 3.13488813947697232335197629858, 3.94579918076548362111975035057, 5.79413736096819793170551158263, 6.03178310616356836964693964274, 7.05977632213055431396055662264, 8.118610472669692296181108858854, 9.130504304216525929211584731319, 9.742374550570625353844875447199, 10.68682439115387352423460166421