| L(s) = 1 | + 2·3-s − 2·5-s + 2·7-s − 3·9-s − 2·11-s − 4·15-s + 4·17-s − 2·19-s + 4·21-s − 2·23-s + 3·25-s − 14·27-s + 2·29-s − 4·31-s − 4·33-s − 4·35-s − 2·37-s − 2·41-s − 10·43-s + 6·45-s + 2·47-s − 8·49-s + 8·51-s − 8·53-s + 4·55-s − 4·57-s − 2·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.755·7-s − 9-s − 0.603·11-s − 1.03·15-s + 0.970·17-s − 0.458·19-s + 0.872·21-s − 0.417·23-s + 3/5·25-s − 2.69·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s − 0.676·35-s − 0.328·37-s − 0.312·41-s − 1.52·43-s + 0.894·45-s + 0.291·47-s − 8/7·49-s + 1.12·51-s − 1.09·53-s + 0.539·55-s − 0.529·57-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13542400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13542400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 72 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 71 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 108 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 68 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 28 T + 339 T^{2} + 28 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 18 T + 184 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 120 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−8.168337968872671493844145148951, −8.140229103645602509662231301235, −7.70892620601337013979370877007, −7.62559048029944466899067760961, −6.90722462804769676410552942144, −6.63419277371911174391962260254, −6.06485208050605258857413127905, −5.62704843522454210109677284487, −5.25384610666469510982419332807, −5.06613338185096932979391348383, −4.26050578600970031803516595451, −4.18712707851061801399109424988, −3.37666690791159365347927213654, −3.33634618229777012030772923309, −2.72809090437138702307806593201, −2.55794171787847280934607824248, −1.60614036440783641206313733085, −1.50348538454980255308313989893, 0, 0,
1.50348538454980255308313989893, 1.60614036440783641206313733085, 2.55794171787847280934607824248, 2.72809090437138702307806593201, 3.33634618229777012030772923309, 3.37666690791159365347927213654, 4.18712707851061801399109424988, 4.26050578600970031803516595451, 5.06613338185096932979391348383, 5.25384610666469510982419332807, 5.62704843522454210109677284487, 6.06485208050605258857413127905, 6.63419277371911174391962260254, 6.90722462804769676410552942144, 7.62559048029944466899067760961, 7.70892620601337013979370877007, 8.140229103645602509662231301235, 8.168337968872671493844145148951
