| L(s) = 1 | + 4·5-s − 2·7-s + 5·11-s + 2·13-s + 6·17-s − 2·19-s + 6·23-s + 5·25-s − 2·29-s − 4·31-s − 8·35-s − 16·37-s + 41-s − 7·43-s − 2·47-s + 7·49-s + 8·53-s + 20·55-s − 5·59-s + 8·65-s − 13·67-s + 16·71-s + 6·73-s − 10·77-s + 8·79-s − 12·83-s + 24·85-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.755·7-s + 1.50·11-s + 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s + 25-s − 0.371·29-s − 0.718·31-s − 1.35·35-s − 2.63·37-s + 0.156·41-s − 1.06·43-s − 0.291·47-s + 49-s + 1.09·53-s + 2.69·55-s − 0.650·59-s + 0.992·65-s − 1.58·67-s + 1.89·71-s + 0.702·73-s − 1.13·77-s + 0.900·79-s − 1.31·83-s + 2.60·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 746496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.197723028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.197723028\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T - 40 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 7 T + 6 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−10.38016479341323186450021173815, −10.04182656772587724229844475071, −9.465739342150552239219420179480, −9.076752826957541386120193771066, −8.945848211396252163943350247444, −8.578359924860229690609078581517, −7.68377632115047883804001126825, −7.36415557397249395309778182779, −6.66102425193647582214758345342, −6.50864501149247461424921089327, −6.13892645052954953804508974511, −5.42858096037832317312738873693, −5.41742110606744453734694803234, −4.71111885715002051973129873200, −3.74027243782593038455786808861, −3.56462638989247576135110153244, −3.03030657342701107608197841606, −2.01426221908549475756113576430, −1.72938968032600947888624752631, −0.931050050523385830925994661487,
0.931050050523385830925994661487, 1.72938968032600947888624752631, 2.01426221908549475756113576430, 3.03030657342701107608197841606, 3.56462638989247576135110153244, 3.74027243782593038455786808861, 4.71111885715002051973129873200, 5.41742110606744453734694803234, 5.42858096037832317312738873693, 6.13892645052954953804508974511, 6.50864501149247461424921089327, 6.66102425193647582214758345342, 7.36415557397249395309778182779, 7.68377632115047883804001126825, 8.578359924860229690609078581517, 8.945848211396252163943350247444, 9.076752826957541386120193771066, 9.465739342150552239219420179480, 10.04182656772587724229844475071, 10.38016479341323186450021173815
