| L(s) = 1 | + 3·5-s − 4·7-s + 2·11-s − 6·13-s + 3·17-s + 19-s + 4·23-s + 4·25-s + 8·29-s − 5·31-s − 12·35-s − 8·37-s − 12·41-s + 7·43-s − 2·47-s + 9·49-s + 11·53-s + 6·55-s − 4·59-s − 6·61-s − 18·65-s − 13·67-s − 5·71-s − 2·73-s − 8·77-s − 79-s + 9·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 1.51·7-s + 0.603·11-s − 1.66·13-s + 0.727·17-s + 0.229·19-s + 0.834·23-s + 4/5·25-s + 1.48·29-s − 0.898·31-s − 2.02·35-s − 1.31·37-s − 1.87·41-s + 1.06·43-s − 0.291·47-s + 9/7·49-s + 1.51·53-s + 0.809·55-s − 0.520·59-s − 0.768·61-s − 2.23·65-s − 1.58·67-s − 0.593·71-s − 0.234·73-s − 0.911·77-s − 0.112·79-s + 0.976·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11916 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 331 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.80481949446969, −16.30766595814990, −15.48804165459190, −14.95199056778020, −14.23928559379523, −13.89328171398169, −13.24573880099985, −12.73193087707620, −12.13309972338637, −11.81023447035421, −10.49073613705640, −10.20012112269420, −9.816203735577466, −9.072347694764842, −8.900544417290191, −7.610363657981263, −6.992749419674753, −6.575442860177626, −5.858889290820570, −5.290879196510531, −4.611122985576726, −3.458029352164703, −2.947720367415092, −2.200676260881047, −1.242270832207598, 0,
1.242270832207598, 2.200676260881047, 2.947720367415092, 3.458029352164703, 4.611122985576726, 5.290879196510531, 5.858889290820570, 6.575442860177626, 6.992749419674753, 7.610363657981263, 8.900544417290191, 9.072347694764842, 9.816203735577466, 10.20012112269420, 10.49073613705640, 11.81023447035421, 12.13309972338637, 12.73193087707620, 13.24573880099985, 13.89328171398169, 14.23928559379523, 14.95199056778020, 15.48804165459190, 16.30766595814990, 16.80481949446969
