| L(s) = 1 | + 2·7-s + 11-s − 8·13-s + 5·17-s + 2·19-s − 8·23-s − 5·25-s + 11·29-s − 7·31-s − 6·37-s + 41-s + 4·43-s − 14·47-s + 3·49-s − 3·53-s + 8·59-s − 12·61-s − 9·67-s + 10·71-s − 7·73-s + 2·77-s + 25·83-s − 6·89-s − 16·91-s − 12·97-s − 2·101-s + 12·103-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s − 2.21·13-s + 1.21·17-s + 0.458·19-s − 1.66·23-s − 25-s + 2.04·29-s − 1.25·31-s − 0.986·37-s + 0.156·41-s + 0.609·43-s − 2.04·47-s + 3/7·49-s − 0.412·53-s + 1.04·59-s − 1.53·61-s − 1.09·67-s + 1.18·71-s − 0.819·73-s + 0.227·77-s + 2.74·83-s − 0.635·89-s − 1.67·91-s − 1.21·97-s − 0.199·101-s + 1.18·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91699776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 37 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 5 T + 29 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 3 p T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 73 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 19 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 97 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 89 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 12 T + 113 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9 T + 153 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 87 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 7 T + 147 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 138 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 25 T + 321 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 182 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 225 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.49518463384154262959322128698, −7.35618368413859942922014567116, −6.90701745794455051420361863499, −6.52492701833764503946545740076, −5.98535997228150715919125209529, −5.96909335183284681638111578499, −5.31188770468733093912461425437, −5.10069711268777047485355102370, −4.72229861920770328497778172301, −4.63210974278375669656928876544, −3.87566691652684810545411004986, −3.73810308657856849559519622533, −3.24345913038008350756599188492, −2.77597498803856012760135702109, −2.22674714033674305834178525694, −2.14026425352790572523620344995, −1.38946216191953009008290482688, −1.16489822613046566498168544389, 0, 0,
1.16489822613046566498168544389, 1.38946216191953009008290482688, 2.14026425352790572523620344995, 2.22674714033674305834178525694, 2.77597498803856012760135702109, 3.24345913038008350756599188492, 3.73810308657856849559519622533, 3.87566691652684810545411004986, 4.63210974278375669656928876544, 4.72229861920770328497778172301, 5.10069711268777047485355102370, 5.31188770468733093912461425437, 5.96909335183284681638111578499, 5.98535997228150715919125209529, 6.52492701833764503946545740076, 6.90701745794455051420361863499, 7.35618368413859942922014567116, 7.49518463384154262959322128698
