| L(s) = 1 | + 14·19-s + 7·25-s + 12·29-s + 10·31-s − 10·37-s − 6·47-s + 18·53-s − 18·59-s − 24·83-s − 2·103-s + 22·109-s − 12·113-s + 19·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 3.21·19-s + 7/5·25-s + 2.22·29-s + 1.79·31-s − 1.64·37-s − 0.875·47-s + 2.47·53-s − 2.34·59-s − 2.63·83-s − 0.197·103-s + 2.10·109-s − 1.12·113-s + 1.72·121-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 + 0.0773·167-s + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49787136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.159978817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.159978817\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 19 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 107 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−8.068050636733633024146880881022, −7.83781235944996863053626419440, −7.26658465646809936746860947552, −7.01526660988118157663818510466, −6.94454934441386724143771326927, −6.39912989782233431593518184315, −6.01268614191039178213110261038, −5.51212847533016117050883062159, −5.40806085644678978779470107830, −4.83535797889663773223953186474, −4.58109066689842512292964496460, −4.40825443130862067097992602740, −3.57906902433984019215726583318, −3.13158175221822986614682777890, −3.11210540632863737282327953495, −2.70571482846972073937925298183, −2.06258198811848863366898403462, −1.24647579997483775547554209837, −1.14964101511043087545773266455, −0.58340925005011158905411957682,
0.58340925005011158905411957682, 1.14964101511043087545773266455, 1.24647579997483775547554209837, 2.06258198811848863366898403462, 2.70571482846972073937925298183, 3.11210540632863737282327953495, 3.13158175221822986614682777890, 3.57906902433984019215726583318, 4.40825443130862067097992602740, 4.58109066689842512292964496460, 4.83535797889663773223953186474, 5.40806085644678978779470107830, 5.51212847533016117050883062159, 6.01268614191039178213110261038, 6.39912989782233431593518184315, 6.94454934441386724143771326927, 7.01526660988118157663818510466, 7.26658465646809936746860947552, 7.83781235944996863053626419440, 8.068050636733633024146880881022
