| L(s) = 1 | + 2-s + 4·4-s + 9·5-s − 7·7-s + 11·8-s + 9·10-s − 11·11-s − 7·14-s + 11·16-s − 42·17-s − 6·19-s + 36·20-s − 11·22-s + 28·23-s + 29·25-s − 28·28-s − 50·29-s − 57·31-s + 44·32-s − 42·34-s − 63·35-s + 58·37-s − 6·38-s + 99·40-s + 52·43-s − 44·44-s + 28·46-s + ⋯ |
| L(s) = 1 | + 1/2·2-s + 4-s + 9/5·5-s − 7-s + 11/8·8-s + 9/10·10-s − 11-s − 1/2·14-s + 0.687·16-s − 2.47·17-s − 0.315·19-s + 9/5·20-s − 1/2·22-s + 1.21·23-s + 1.15·25-s − 28-s − 1.72·29-s − 1.83·31-s + 11/8·32-s − 1.23·34-s − 9/5·35-s + 1.56·37-s − 0.157·38-s + 2.47·40-s + 1.20·43-s − 44-s + 0.608·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.359902009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.359902009\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - 3 T^{2} - p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 - p T + p^{2} T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 42 T + 877 T^{2} + 42 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 373 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 28 T + 255 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 25 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 57 T + 2044 T^{2} + 57 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 58 T + 1995 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3350 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 132 T + 8017 T^{2} - 132 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31 T - 1848 T^{2} - 31 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 3556 T^{2} - 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 24 T + 3913 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 52 T - 1785 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 64 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 12 T + 5377 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 17 T - 5952 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10895 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 138 T + 14269 T^{2} - 138 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 10391 T^{2} + p^{4} T^{4} \) |
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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
−15.06555296587098876195445934958, −14.36365247883001466408403729998, −13.52954398153912899098588555499, −13.39620650067790784591216457611, −12.89299345091809083344005955286, −12.75048928044372332479797124958, −11.34482983251737025809429236160, −11.00838277679250217845286392601, −10.59038683511004238360419950412, −9.951760754909893368765408462828, −9.142130163369524531045877238218, −8.948778908868991135482103589218, −7.31572712477966137367147373226, −7.26747838476302701236199733169, −6.21556292057779321661440295070, −5.91002149264607674439617771390, −5.04973741063359132101911293918, −4.03625502440685695959051237746, −2.46048518333911412747807935888, −2.17109950964999385061777930053,
2.17109950964999385061777930053, 2.46048518333911412747807935888, 4.03625502440685695959051237746, 5.04973741063359132101911293918, 5.91002149264607674439617771390, 6.21556292057779321661440295070, 7.26747838476302701236199733169, 7.31572712477966137367147373226, 8.948778908868991135482103589218, 9.142130163369524531045877238218, 9.951760754909893368765408462828, 10.59038683511004238360419950412, 11.00838277679250217845286392601, 11.34482983251737025809429236160, 12.75048928044372332479797124958, 12.89299345091809083344005955286, 13.39620650067790784591216457611, 13.52954398153912899098588555499, 14.36365247883001466408403729998, 15.06555296587098876195445934958
