L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 5-s + (0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 0.999·12-s + 13-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)17-s − 0.999·18-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s − 5-s + (0.499 − 0.866i)6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s + 0.999·12-s + 13-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (1 − 1.73i)17-s − 0.999·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7559562603\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7559562603\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 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
−9.123490154657383279849162407230, −8.352971208711085777644249772539, −7.72631677512436013508027944563, −7.22218939015665074314621385008, −6.25395447626052788118103128782, −5.59166644393434739119604911868, −4.75654329776446243524431860871, −3.63596889452085221097128607364, −2.74892321565279398541930181665, −0.58333095216195354247331909647,
1.47940075909707004064507090842, 3.16942435994920246102566525812, 3.86577231823986330470112004461, 4.40237697517048307052104126073, 5.52104663999998049994575977915, 6.02318342050676670170937782801, 7.36272270622419921711122879124, 8.344047369398747222890570315653, 9.168553505774915029795332745985, 9.974784129365682638359242953395