L(s) = 1 | + (−0.809 − 1.40i)2-s + (0.809 + 1.40i)3-s + (−0.809 + 1.40i)4-s + (1.30 − 2.26i)6-s + (0.309 − 0.535i)7-s + 0.999·8-s + (−0.809 + 1.40i)9-s + (0.5 + 0.866i)11-s − 2.61·12-s − 14-s + 2.61·18-s + (−0.809 + 1.40i)19-s + 21-s + (0.809 − 1.40i)22-s + (−0.309 − 0.535i)23-s + (0.809 + 1.40i)24-s + ⋯ |
L(s) = 1 | + (−0.809 − 1.40i)2-s + (0.809 + 1.40i)3-s + (−0.809 + 1.40i)4-s + (1.30 − 2.26i)6-s + (0.309 − 0.535i)7-s + 0.999·8-s + (−0.809 + 1.40i)9-s + (0.5 + 0.866i)11-s − 2.61·12-s − 14-s + 2.61·18-s + (−0.809 + 1.40i)19-s + 21-s + (0.809 − 1.40i)22-s + (−0.309 − 0.535i)23-s + (0.809 + 1.40i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9420899148\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9420899148\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - 1.61T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.618T + T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−9.751437454139908517872216840112, −8.933895856850619119019741596594, −8.372314642148071192982909755015, −7.63603151567236868656639011908, −6.29443076873478273602733820603, −4.78816060941540580756413394176, −4.16613542139994979735005208055, −3.52785419925560541736864567178, −2.56247211382127301731088944395, −1.56014463873826563548800012018,
0.897805846323087410129088445687, 2.19796428821317423964597392643, 3.26144336822844253568526095690, 4.87338470015644872633230847770, 5.90687373248853390396662274767, 6.53887047486827115583102688257, 7.10476122860471875531427344265, 7.84025876069348188935555164022, 8.581750714540021898692226562912, 8.814501259102035688436781298781