| L(s) = 1 | − 3·4-s + 4·5-s − 4·11-s + 5·16-s − 12·20-s + 2·25-s + 12·37-s + 12·44-s + 49-s − 12·53-s − 16·55-s − 24·59-s − 3·64-s + 8·67-s + 20·80-s + 28·89-s + 36·97-s − 6·100-s + 16·103-s + 28·113-s + 5·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s − 36·148-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1.78·5-s − 1.20·11-s + 5/4·16-s − 2.68·20-s + 2/5·25-s + 1.97·37-s + 1.80·44-s + 1/7·49-s − 1.64·53-s − 2.15·55-s − 3.12·59-s − 3/8·64-s + 0.977·67-s + 2.23·80-s + 2.96·89-s + 3.65·97-s − 3/5·100-s + 1.57·103-s + 2.63·113-s + 5/11·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.95·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480249 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.386271862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.386271862\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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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
−8.783200717465470995950515779184, −7.976525779912433683772333851675, −7.81295430367747747098376616892, −7.39398850413165927730875614743, −6.34518093653275179095390844834, −6.01581061696917710851209529028, −5.94607665445287604157300008429, −5.02709416296405066539370067618, −4.92403553365775964167265796866, −4.46792640657225865455095535714, −3.62088779784825026995011416777, −3.05422074105458389777226041971, −2.28059162307602342837183603744, −1.77611296934943774735876447029, −0.63334663369662260021570161928,
0.63334663369662260021570161928, 1.77611296934943774735876447029, 2.28059162307602342837183603744, 3.05422074105458389777226041971, 3.62088779784825026995011416777, 4.46792640657225865455095535714, 4.92403553365775964167265796866, 5.02709416296405066539370067618, 5.94607665445287604157300008429, 6.01581061696917710851209529028, 6.34518093653275179095390844834, 7.39398850413165927730875614743, 7.81295430367747747098376616892, 7.976525779912433683772333851675, 8.783200717465470995950515779184
