| L(s) = 1 | − 2·2-s − 4·3-s + 2·4-s − 4·5-s + 8·6-s − 5·7-s + 7·9-s + 8·10-s − 11-s − 8·12-s − 5·13-s + 10·14-s + 16·15-s − 4·16-s − 3·17-s − 14·18-s − 2·19-s − 8·20-s + 20·21-s + 2·22-s − 3·23-s + 7·25-s + 10·26-s − 4·27-s − 10·28-s + 9·29-s − 32·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 2.30·3-s + 4-s − 1.78·5-s + 3.26·6-s − 1.88·7-s + 7/3·9-s + 2.52·10-s − 0.301·11-s − 2.30·12-s − 1.38·13-s + 2.67·14-s + 4.13·15-s − 16-s − 0.727·17-s − 3.29·18-s − 0.458·19-s − 1.78·20-s + 4.36·21-s + 0.426·22-s − 0.625·23-s + 7/5·25-s + 1.96·26-s − 0.769·27-s − 1.88·28-s + 1.67·29-s − 5.84·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7004 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7004 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 103 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 9 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 23 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T - 31 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T - 39 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 65 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 11 T + 126 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 19 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 11 T + 133 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 18 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 136 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−17.5205853763, −16.9666984659, −16.5258731157, −16.3406153566, −15.9566116209, −15.5219984059, −14.9940938180, −13.9945313394, −12.9749440075, −12.7022770611, −12.0232872098, −11.8389588957, −11.2475443958, −10.7860019718, −10.1856288760, −9.88382233178, −9.07076087621, −8.35915997083, −7.53269907197, −7.15360615398, −6.41181326759, −6.09688502751, −4.83554991404, −4.38874448694, −3.02123738745, 0, 0,
3.02123738745, 4.38874448694, 4.83554991404, 6.09688502751, 6.41181326759, 7.15360615398, 7.53269907197, 8.35915997083, 9.07076087621, 9.88382233178, 10.1856288760, 10.7860019718, 11.2475443958, 11.8389588957, 12.0232872098, 12.7022770611, 12.9749440075, 13.9945313394, 14.9940938180, 15.5219984059, 15.9566116209, 16.3406153566, 16.5258731157, 16.9666984659, 17.5205853763
