L(s) = 1 | + 3-s + 7-s + 11-s − 17-s + 21-s − 2·23-s + 25-s − 27-s − 29-s + 31-s + 33-s − 37-s + 2·41-s − 51-s + 59-s − 61-s − 2·69-s + 75-s + 77-s − 81-s − 83-s − 87-s + 93-s − 109-s − 111-s − 113-s − 119-s + ⋯ |
L(s) = 1 | + 3-s + 7-s + 11-s − 17-s + 21-s − 2·23-s + 25-s − 27-s − 29-s + 31-s + 33-s − 37-s + 2·41-s − 51-s + 59-s − 61-s − 2·69-s + 75-s + 77-s − 81-s − 83-s − 87-s + 93-s − 109-s − 111-s − 113-s − 119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.585022164\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.585022164\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 83 | \( 1 + T \) |
good | 3 | \( 1 - T + T^{2} \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 - T + T^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( ( 1 + T )^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( ( 1 - T )^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( 1 - T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( ( 1 - T )( 1 + T ) \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.559465075735668886731535413435, −8.918781378133019943236457369013, −8.287123648012125383867424831268, −7.63886962710502613850075455603, −6.61793116824543450857127683638, −5.68312089006156878812846942551, −4.46864943192264282147295107706, −3.82511842162550708535569567788, −2.56236355163019656337205767838, −1.67036991833096906207723721274,
1.67036991833096906207723721274, 2.56236355163019656337205767838, 3.82511842162550708535569567788, 4.46864943192264282147295107706, 5.68312089006156878812846942551, 6.61793116824543450857127683638, 7.63886962710502613850075455603, 8.287123648012125383867424831268, 8.918781378133019943236457369013, 9.559465075735668886731535413435