| L(s) = 1 | − i·3-s + 4-s − i·7-s − 9-s − i·12-s − 13-s + 16-s + 19-s − 21-s − 25-s + i·27-s − i·28-s + i·31-s − 36-s + i·37-s + ⋯ |
| L(s) = 1 | − i·3-s + 4-s − i·7-s − 9-s − i·12-s − 13-s + 16-s + 19-s − 21-s − 25-s + i·27-s − i·28-s + i·31-s − 36-s + i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.160425197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160425197\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - T^{2} \) |
| 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - iT - T^{2} \) |
| 37 | \( 1 - iT - T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + iT - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + 2iT - T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - iT - T^{2} \) |
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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
−10.33767084617862649312571636417, −9.436042633965790400104060375776, −8.082946923644960185747051369010, −7.47840772732253939270171255021, −6.95648980062904258100146040314, −6.08873293668499679089115136713, −5.04471798888127980928689362369, −3.51199453032489949428313646278, −2.48269598618486767460476972861, −1.28083172523360863352780036711,
2.23116220643134679818699238569, 3.00651133261333091032583759329, 4.23371768031106467026078435975, 5.52547231356637765662424170241, 5.85362616513724557997568557462, 7.20775196127926524992927847255, 7.985556402735614193597198120505, 9.094861609146242046503673493876, 9.724572920709902619385376824533, 10.48063571240119516119357018170
