| L(s) = 1 | + 5-s − 6·11-s + 17-s − 3·19-s + 7·23-s − 4·25-s − 6·29-s − 8·31-s + 7·37-s + 9·43-s + 6·47-s − 2·53-s − 6·55-s + 11·59-s + 6·61-s − 9·67-s − 9·71-s + 12·73-s + 8·79-s − 4·83-s + 85-s + 5·89-s − 3·95-s + 4·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.80·11-s + 0.242·17-s − 0.688·19-s + 1.45·23-s − 4/5·25-s − 1.11·29-s − 1.43·31-s + 1.15·37-s + 1.37·43-s + 0.875·47-s − 0.274·53-s − 0.809·55-s + 1.43·59-s + 0.768·61-s − 1.09·67-s − 1.06·71-s + 1.40·73-s + 0.900·79-s − 0.439·83-s + 0.108·85-s + 0.529·89-s − 0.307·95-s + 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−13.59492066310005, −13.24361289078125, −12.96376113214314, −12.54918284730068, −11.93366777103884, −11.09122396427117, −10.94718091441179, −10.54896561102434, −9.867177751388276, −9.439334654312216, −9.005615460626800, −8.341162758969582, −7.870612902668780, −7.299951680679135, −7.083869192786616, −6.070277811807484, −5.769855987700633, −5.274989819878476, −4.819084010599937, −4.029690938643811, −3.541401830886631, −2.611911863560403, −2.452129127752485, −1.695490290844160, −0.7815674496997250, 0,
0.7815674496997250, 1.695490290844160, 2.452129127752485, 2.611911863560403, 3.541401830886631, 4.029690938643811, 4.819084010599937, 5.274989819878476, 5.769855987700633, 6.070277811807484, 7.083869192786616, 7.299951680679135, 7.870612902668780, 8.341162758969582, 9.005615460626800, 9.439334654312216, 9.867177751388276, 10.54896561102434, 10.94718091441179, 11.09122396427117, 11.93366777103884, 12.54918284730068, 12.96376113214314, 13.24361289078125, 13.59492066310005
