| L(s) = 1 | − 2·7-s − 9-s + 12·23-s + 10·25-s + 16·31-s − 24·41-s + 24·47-s + 3·49-s + 2·63-s − 12·71-s + 20·73-s − 8·79-s + 81-s − 24·89-s − 20·97-s − 16·103-s − 12·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 24·161-s + 163-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1/3·9-s + 2.50·23-s + 2·25-s + 2.87·31-s − 3.74·41-s + 3.50·47-s + 3/7·49-s + 0.251·63-s − 1.42·71-s + 2.34·73-s − 0.900·79-s + 1/9·81-s − 2.54·89-s − 2.03·97-s − 1.57·103-s − 1.12·113-s − 1.27·121-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 − 1.89·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28901376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.398744519\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.398744519\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.741376925226365980698439582215, −8.197451565090073380256223845883, −7.59675164895588334070348386317, −7.07160129917770707532353241715, −6.87401811512946843816273819919, −6.59254268668578288248080930839, −6.52019958098699397443596794088, −5.76968604103250793534544153391, −5.27091455477345115090835253257, −5.25626809348525549333895544894, −4.79288320765838573004329877568, −4.18499098222412554668901955667, −4.07276950725577530288946453273, −3.15676367888424551467572689747, −3.10707592291113895267410181114, −2.71297757063755141625942169239, −2.41287830004445153178061741925, −1.26984996622580969784215597678, −1.20854458323223828249294801970, −0.46189172833971569665476868801,
0.46189172833971569665476868801, 1.20854458323223828249294801970, 1.26984996622580969784215597678, 2.41287830004445153178061741925, 2.71297757063755141625942169239, 3.10707592291113895267410181114, 3.15676367888424551467572689747, 4.07276950725577530288946453273, 4.18499098222412554668901955667, 4.79288320765838573004329877568, 5.25626809348525549333895544894, 5.27091455477345115090835253257, 5.76968604103250793534544153391, 6.52019958098699397443596794088, 6.59254268668578288248080930839, 6.87401811512946843816273819919, 7.07160129917770707532353241715, 7.59675164895588334070348386317, 8.197451565090073380256223845883, 8.741376925226365980698439582215
