| L(s) = 1 | + 2·2-s + 2·4-s − 2·5-s − 5·7-s + 4·8-s − 4·10-s − 2·11-s + 2·13-s − 10·14-s + 8·16-s − 19-s − 4·20-s − 4·22-s + 5·25-s + 4·26-s − 10·28-s − 8·29-s − 9·31-s + 8·32-s + 10·35-s − 3·37-s − 2·38-s − 8·40-s + 20·41-s + 10·43-s − 4·44-s − 6·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.894·5-s − 1.88·7-s + 1.41·8-s − 1.26·10-s − 0.603·11-s + 0.554·13-s − 2.67·14-s + 2·16-s − 0.229·19-s − 0.894·20-s − 0.852·22-s + 25-s + 0.784·26-s − 1.88·28-s − 1.48·29-s − 1.61·31-s + 1.41·32-s + 1.69·35-s − 0.493·37-s − 0.324·38-s − 1.26·40-s + 3.12·41-s + 1.52·43-s − 0.603·44-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.228868795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228868795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 3 T - 64 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−15.36560536854009396311158063456, −14.50700982969809282108683898068, −14.20505102814409591011598352646, −13.35326816699018380033904999717, −13.08004825465329500396874042276, −12.56274054701654781931107198693, −12.45412352413842447634937223453, −11.38026040487639080475052708788, −10.66730152363417441123708123421, −10.62667050589887060310804297029, −9.390691972747526801245265580536, −9.081367891683529429878694495088, −7.72619710623844035149662313582, −7.51773080171603975996014291575, −6.65467300645858647330142044042, −5.86931472424151875016756922329, −5.27089321981726813638725122212, −3.96149457595243024167394929240, −3.88826097130657345122151487621, −2.77247219492568790052505064510,
2.77247219492568790052505064510, 3.88826097130657345122151487621, 3.96149457595243024167394929240, 5.27089321981726813638725122212, 5.86931472424151875016756922329, 6.65467300645858647330142044042, 7.51773080171603975996014291575, 7.72619710623844035149662313582, 9.081367891683529429878694495088, 9.390691972747526801245265580536, 10.62667050589887060310804297029, 10.66730152363417441123708123421, 11.38026040487639080475052708788, 12.45412352413842447634937223453, 12.56274054701654781931107198693, 13.08004825465329500396874042276, 13.35326816699018380033904999717, 14.20505102814409591011598352646, 14.50700982969809282108683898068, 15.36560536854009396311158063456
