| L(s) = 1 | + 3·4-s − 2·11-s + 5·16-s − 12·29-s − 16·31-s + 4·41-s − 6·44-s − 2·49-s − 8·59-s + 12·61-s + 3·64-s + 8·79-s − 12·89-s − 4·101-s + 4·109-s − 36·116-s + 3·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 12·164-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.603·11-s + 5/4·16-s − 2.22·29-s − 2.87·31-s + 0.624·41-s − 0.904·44-s − 2/7·49-s − 1.04·59-s + 1.53·61-s + 3/8·64-s + 0.900·79-s − 1.27·89-s − 0.398·101-s + 0.383·109-s − 3.34·116-s + 3/11·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.937·164-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.383000754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.383000754\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−9.313090476492710248391318116837, −8.572118508735431150178297917237, −8.426414204575432439515965727940, −7.65425551010714806031984579471, −7.52416623741558427761001439276, −7.39237206538559556034207148451, −6.76075008149238254837919490492, −6.62739717935609349818089283314, −5.81035351536980736913727158654, −5.79021730882807693922797072594, −5.35436801606081535945364348358, −4.96266235526207981575931562842, −4.05335386818013800255305889111, −3.96787474701537084908378451881, −3.18560476081113685080029332595, −3.05111798410349557260718715854, −2.22233493526386804438013593533, −1.93158060445720587440926283626, −1.59218125761055456862938865920, −0.46773758599670833236951919248,
0.46773758599670833236951919248, 1.59218125761055456862938865920, 1.93158060445720587440926283626, 2.22233493526386804438013593533, 3.05111798410349557260718715854, 3.18560476081113685080029332595, 3.96787474701537084908378451881, 4.05335386818013800255305889111, 4.96266235526207981575931562842, 5.35436801606081535945364348358, 5.79021730882807693922797072594, 5.81035351536980736913727158654, 6.62739717935609349818089283314, 6.76075008149238254837919490492, 7.39237206538559556034207148451, 7.52416623741558427761001439276, 7.65425551010714806031984579471, 8.426414204575432439515965727940, 8.572118508735431150178297917237, 9.313090476492710248391318116837
