| L(s) = 1 | + 3·3-s + 5-s − 7-s + 3·9-s − 2·11-s + 3·15-s + 4·17-s − 6·19-s − 3·21-s + 3·23-s + 18·29-s − 4·31-s − 6·33-s − 35-s + 4·37-s − 14·41-s + 10·43-s + 3·45-s + 8·47-s − 6·49-s + 12·51-s + 2·53-s − 2·55-s − 18·57-s + 10·59-s − 61-s − 3·63-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 0.377·7-s + 9-s − 0.603·11-s + 0.774·15-s + 0.970·17-s − 1.37·19-s − 0.654·21-s + 0.625·23-s + 3.34·29-s − 0.718·31-s − 1.04·33-s − 0.169·35-s + 0.657·37-s − 2.18·41-s + 1.52·43-s + 0.447·45-s + 1.16·47-s − 6/7·49-s + 1.68·51-s + 0.274·53-s − 0.269·55-s − 2.38·57-s + 1.30·59-s − 0.128·61-s − 0.377·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.351740693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.351740693\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 9 T + 14 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + T - 88 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 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
−10.54595906551316230320015450579, −10.44313820110958323467439588754, −10.22414736738431462758807453752, −9.508081184284812214444213044725, −9.179944107682140422496415761408, −8.598856385245678333034140305252, −8.538939283165961772756463139182, −8.021173760011497715342740934621, −7.59088936012645989924301796270, −7.02546501483875308751522157258, −6.41220196592293416282032508898, −6.15131089195191913600287833980, −5.36692613556797775986044857335, −4.81677520012078906666133574965, −4.31878144903806376798443402826, −3.34892418101635062151565630805, −3.28033937220239823772823810976, −2.40551751294894912308068564223, −2.27698055848917095238064559182, −0.995348967233099755643506060634,
0.995348967233099755643506060634, 2.27698055848917095238064559182, 2.40551751294894912308068564223, 3.28033937220239823772823810976, 3.34892418101635062151565630805, 4.31878144903806376798443402826, 4.81677520012078906666133574965, 5.36692613556797775986044857335, 6.15131089195191913600287833980, 6.41220196592293416282032508898, 7.02546501483875308751522157258, 7.59088936012645989924301796270, 8.021173760011497715342740934621, 8.538939283165961772756463139182, 8.598856385245678333034140305252, 9.179944107682140422496415761408, 9.508081184284812214444213044725, 10.22414736738431462758807453752, 10.44313820110958323467439588754, 10.54595906551316230320015450579
