| L(s) = 1 | − 4-s − 2·11-s + 16-s − 10·19-s + 10·29-s − 6·31-s − 4·41-s + 2·44-s + 5·49-s − 20·59-s + 14·61-s − 64-s − 14·71-s + 10·76-s − 20·79-s − 30·89-s − 4·101-s + 20·109-s − 10·116-s + 3·121-s + 6·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.603·11-s + 1/4·16-s − 2.29·19-s + 1.85·29-s − 1.07·31-s − 0.624·41-s + 0.301·44-s + 5/7·49-s − 2.60·59-s + 1.79·61-s − 1/8·64-s − 1.66·71-s + 1.14·76-s − 2.25·79-s − 3.17·89-s − 0.398·101-s + 1.91·109-s − 0.928·116-s + 3/11·121-s + 0.538·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2464471927\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2464471927\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 105 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
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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
−8.599111367044997743709252613469, −8.293497056456777197522810281609, −7.77698568717685157876604133156, −7.31035536703054171355992257290, −6.99857713958959027020905124933, −6.66721968101418712708068837256, −6.27743327428404554048365854275, −5.72670472362690878459859261829, −5.69878268637506795002291453962, −5.12681906756756827202537950668, −4.54791920175645759252507602687, −4.34891174660402439252994556939, −4.23786346101599439942700457805, −3.46923344538737046223967797790, −3.08740013282022131563673577923, −2.66112295313218170371540037646, −2.17385223832909538997528907614, −1.66953200584911555653938587152, −1.08793348431989856673173371178, −0.14342871787182213213477064196,
0.14342871787182213213477064196, 1.08793348431989856673173371178, 1.66953200584911555653938587152, 2.17385223832909538997528907614, 2.66112295313218170371540037646, 3.08740013282022131563673577923, 3.46923344538737046223967797790, 4.23786346101599439942700457805, 4.34891174660402439252994556939, 4.54791920175645759252507602687, 5.12681906756756827202537950668, 5.69878268637506795002291453962, 5.72670472362690878459859261829, 6.27743327428404554048365854275, 6.66721968101418712708068837256, 6.99857713958959027020905124933, 7.31035536703054171355992257290, 7.77698568717685157876604133156, 8.293497056456777197522810281609, 8.599111367044997743709252613469
