| L(s) = 1 | + 5-s − 7-s + 4·11-s − 10·13-s + 6·17-s + 5·19-s + 2·23-s + 4·29-s + 9·31-s − 35-s + 11·37-s + 16·41-s + 2·43-s + 6·47-s − 6·49-s − 4·53-s + 4·55-s + 6·59-s + 14·61-s − 10·65-s + 67-s − 24·71-s − 11·73-s − 4·77-s − 79-s + 8·83-s + 6·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.20·11-s − 2.77·13-s + 1.45·17-s + 1.14·19-s + 0.417·23-s + 0.742·29-s + 1.61·31-s − 0.169·35-s + 1.80·37-s + 2.49·41-s + 0.304·43-s + 0.875·47-s − 6/7·49-s − 0.549·53-s + 0.539·55-s + 0.781·59-s + 1.79·61-s − 1.24·65-s + 0.122·67-s − 2.84·71-s − 1.28·73-s − 0.455·77-s − 0.112·79-s + 0.878·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6350400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.289280315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.289280315\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 - 4 T + 5 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 2 T - 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−9.273162377825351207644446445699, −8.945500486712343537465878625794, −8.130786465887345410829399055650, −7.916401975286515566285235869908, −7.59052378947856956372143734361, −7.22968603949797022611143004852, −6.72042549731724403430717921221, −6.60829075546506675486776675896, −5.84349386158262039328615542371, −5.64795309757959874128509877582, −5.30497190873949320518899625233, −4.67822718146505756568354479793, −4.22004423089725902282729164698, −4.21192247372831213276148680650, −3.04635900313064856331066064346, −2.95557224206179380213546575652, −2.60863354991612477251737618721, −1.90476273005603457859720657343, −1.03107015366364410136875843680, −0.74695568084358125539610465876,
0.74695568084358125539610465876, 1.03107015366364410136875843680, 1.90476273005603457859720657343, 2.60863354991612477251737618721, 2.95557224206179380213546575652, 3.04635900313064856331066064346, 4.21192247372831213276148680650, 4.22004423089725902282729164698, 4.67822718146505756568354479793, 5.30497190873949320518899625233, 5.64795309757959874128509877582, 5.84349386158262039328615542371, 6.60829075546506675486776675896, 6.72042549731724403430717921221, 7.22968603949797022611143004852, 7.59052378947856956372143734361, 7.916401975286515566285235869908, 8.130786465887345410829399055650, 8.945500486712343537465878625794, 9.273162377825351207644446445699
