| L(s) = 1 | − 4-s − 6·11-s + 16-s + 8·19-s − 6·29-s + 10·31-s + 6·44-s + 10·49-s + 24·59-s + 4·61-s − 64-s − 24·71-s − 8·76-s + 26·79-s − 24·89-s − 30·101-s − 40·109-s + 6·116-s + 5·121-s − 10·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.80·11-s + 1/4·16-s + 1.83·19-s − 1.11·29-s + 1.79·31-s + 0.904·44-s + 10/7·49-s + 3.12·59-s + 0.512·61-s − 1/8·64-s − 2.84·71-s − 0.917·76-s + 2.92·79-s − 2.54·89-s − 2.98·101-s − 3.83·109-s + 0.557·116-s + 5/11·121-s − 0.898·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.445180192\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.445180192\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 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
−9.647668444572553716119116250019, −9.495157831553610679814199769671, −9.160288088543018681544662586376, −8.397653175916324563837969921205, −8.146390955941154337500793267583, −7.980300497392279542860719910665, −7.41641084694036046840800316640, −6.92065656097945310407842743974, −6.77783051042951068768569766255, −5.71320591401434983249205438488, −5.64086534872231543205755474175, −5.28761980035629810492378082530, −4.89340176129836874320029489098, −4.04565519395004100741305244654, −4.02848678937883208122259448399, −2.93583972209967448686707042237, −2.91946066844286205900365904418, −2.22970871807828956655362370397, −1.28861144535864305863726083217, −0.53834880395933276273702940229,
0.53834880395933276273702940229, 1.28861144535864305863726083217, 2.22970871807828956655362370397, 2.91946066844286205900365904418, 2.93583972209967448686707042237, 4.02848678937883208122259448399, 4.04565519395004100741305244654, 4.89340176129836874320029489098, 5.28761980035629810492378082530, 5.64086534872231543205755474175, 5.71320591401434983249205438488, 6.77783051042951068768569766255, 6.92065656097945310407842743974, 7.41641084694036046840800316640, 7.980300497392279542860719910665, 8.146390955941154337500793267583, 8.397653175916324563837969921205, 9.160288088543018681544662586376, 9.495157831553610679814199769671, 9.647668444572553716119116250019
