| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 2·5-s − 4·6-s − 4·8-s + 3·9-s + 4·10-s − 4·11-s + 6·12-s − 4·15-s + 5·16-s − 8·17-s − 6·18-s + 8·19-s − 6·20-s + 8·22-s + 12·23-s − 8·24-s + 3·25-s + 4·27-s + 8·29-s + 8·30-s + 12·31-s − 6·32-s − 8·33-s + 16·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 1.41·8-s + 9-s + 1.26·10-s − 1.20·11-s + 1.73·12-s − 1.03·15-s + 5/4·16-s − 1.94·17-s − 1.41·18-s + 1.83·19-s − 1.34·20-s + 1.70·22-s + 2.50·23-s − 1.63·24-s + 3/5·25-s + 0.769·27-s + 1.48·29-s + 1.46·30-s + 2.15·31-s − 1.06·32-s − 1.39·33-s + 2.74·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.767093441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.767093441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 48 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_4$ | \( 1 - 12 T + 74 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 80 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 72 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 80 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 88 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 222 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 174 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.575794259256383376945330298536, −9.359389671984140178074224848775, −8.691713581639359702128889918960, −8.449848927606038986166160857196, −8.255652044158678091899951984206, −8.000189512234428561791894612207, −7.26040578021397072339642465224, −7.16092719728154909945631990735, −6.71138443765284260195545084810, −6.52779371903591659927446663777, −5.37612702172934133440579256508, −5.28602139302763194320045335046, −4.38486779380176509575664886730, −4.36440397428401516971861550638, −3.20166362606442271447288247584, −3.14514599817664623456611992581, −2.43280695826534350118871297251, −2.36523121376459884943110542178, −0.918688926678102251627960425456, −0.881882603000887135653074500524,
0.881882603000887135653074500524, 0.918688926678102251627960425456, 2.36523121376459884943110542178, 2.43280695826534350118871297251, 3.14514599817664623456611992581, 3.20166362606442271447288247584, 4.36440397428401516971861550638, 4.38486779380176509575664886730, 5.28602139302763194320045335046, 5.37612702172934133440579256508, 6.52779371903591659927446663777, 6.71138443765284260195545084810, 7.16092719728154909945631990735, 7.26040578021397072339642465224, 8.000189512234428561791894612207, 8.255652044158678091899951984206, 8.449848927606038986166160857196, 8.691713581639359702128889918960, 9.359389671984140178074224848775, 9.575794259256383376945330298536
