L(s) = 1 | + 8·7-s + 6·9-s + 4·17-s − 8·23-s − 25-s − 16·31-s + 12·41-s + 8·47-s + 34·49-s + 48·63-s + 12·73-s + 27·81-s + 12·89-s − 28·97-s − 8·103-s + 36·113-s + 32·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s − 64·161-s + ⋯ |
L(s) = 1 | + 3.02·7-s + 2·9-s + 0.970·17-s − 1.66·23-s − 1/5·25-s − 2.87·31-s + 1.87·41-s + 1.16·47-s + 34/7·49-s + 6.04·63-s + 1.40·73-s + 3·81-s + 1.27·89-s − 2.84·97-s − 0.788·103-s + 3.38·113-s + 2.93·119-s + 6/11·121-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 + 1.94·153-s + 0.0798·157-s − 5.04·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.406960782\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.406960782\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.783476976910268193793207618104, −9.556883724283425538484619981150, −9.049720005539771589039334819287, −8.601478980000924470268935778253, −7.934998029268371165969628099734, −7.920013100322538625068420066873, −7.45612804385445602217895831863, −7.35788545093417771082675510674, −6.72752498027776395137044191321, −5.98427364107726571581140300943, −5.41392899746999746386039428844, −5.38677145049316631031482791381, −4.60988225711371971150483130425, −4.40106414574905105150304509796, −3.91640263377770793539187093723, −3.60891605358348380574518074359, −2.27093942794656894904473638871, −2.00603431714875057107714298675, −1.50613779696024433616730292323, −1.01509157121117666870023522294,
1.01509157121117666870023522294, 1.50613779696024433616730292323, 2.00603431714875057107714298675, 2.27093942794656894904473638871, 3.60891605358348380574518074359, 3.91640263377770793539187093723, 4.40106414574905105150304509796, 4.60988225711371971150483130425, 5.38677145049316631031482791381, 5.41392899746999746386039428844, 5.98427364107726571581140300943, 6.72752498027776395137044191321, 7.35788545093417771082675510674, 7.45612804385445602217895831863, 7.920013100322538625068420066873, 7.934998029268371165969628099734, 8.601478980000924470268935778253, 9.049720005539771589039334819287, 9.556883724283425538484619981150, 9.783476976910268193793207618104