| L(s) = 1 | − 2-s − 4-s + 3·8-s + 2·9-s − 2·11-s − 16-s − 2·18-s + 2·22-s − 8·23-s + 25-s − 4·29-s − 5·32-s − 2·36-s + 4·37-s + 2·43-s + 2·44-s + 8·46-s − 7·49-s − 50-s − 4·53-s + 4·58-s + 7·64-s − 2·67-s − 12·71-s + 6·72-s − 4·74-s + 8·79-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 2/3·9-s − 0.603·11-s − 1/4·16-s − 0.471·18-s + 0.426·22-s − 1.66·23-s + 1/5·25-s − 0.742·29-s − 0.883·32-s − 1/3·36-s + 0.657·37-s + 0.304·43-s + 0.301·44-s + 1.17·46-s − 49-s − 0.141·50-s − 0.549·53-s + 0.525·58-s + 7/8·64-s − 0.244·67-s − 1.42·71-s + 0.707·72-s − 0.464·74-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
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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
−9.055561680211748766084205841291, −8.543241551126492707286786354706, −8.039021793823657532431972491689, −7.63546022773226113388266796515, −7.39617678033004991975110680715, −6.57301558387128420006927440372, −6.11954490427457680165669328554, −5.41286308799244061524984199768, −4.96236062001387682516756401784, −4.21578467950668468003676949716, −3.97503517295971009107398012721, −3.04200525160127382910160590154, −2.11672424223854521329905169632, −1.36169243264703080367072142361, 0,
1.36169243264703080367072142361, 2.11672424223854521329905169632, 3.04200525160127382910160590154, 3.97503517295971009107398012721, 4.21578467950668468003676949716, 4.96236062001387682516756401784, 5.41286308799244061524984199768, 6.11954490427457680165669328554, 6.57301558387128420006927440372, 7.39617678033004991975110680715, 7.63546022773226113388266796515, 8.039021793823657532431972491689, 8.543241551126492707286786354706, 9.055561680211748766084205841291
