| L(s) = 1 | − 2·5-s − 2·7-s + 8·13-s − 10·17-s + 2·19-s + 8·23-s − 4·25-s + 2·29-s + 4·35-s − 8·37-s − 16·41-s − 4·43-s + 6·47-s + 3·49-s − 6·53-s − 16·59-s + 16·61-s − 16·65-s − 8·67-s + 2·71-s + 8·73-s − 4·79-s − 10·83-s + 20·85-s − 16·91-s − 4·95-s + 12·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 2.21·13-s − 2.42·17-s + 0.458·19-s + 1.66·23-s − 4/5·25-s + 0.371·29-s + 0.676·35-s − 1.31·37-s − 2.49·41-s − 0.609·43-s + 0.875·47-s + 3/7·49-s − 0.824·53-s − 2.08·59-s + 2.04·61-s − 1.98·65-s − 0.977·67-s + 0.237·71-s + 0.936·73-s − 0.450·79-s − 1.09·83-s + 2.16·85-s − 1.67·91-s − 0.410·95-s + 1.21·97-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 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 10 T + 56 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 56 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 2 T + 116 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 164 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−7.32596981093123567795109009990, −7.17127843081953679290490101491, −6.82340138894936012718370614380, −6.52554882937912271436068041833, −6.21961063699938830874031650557, −6.03969934927840819380248966825, −5.30764316957079062099129793117, −5.20940538295844131186310197427, −4.57526632988669604170397183389, −4.45959312612735554465634134030, −3.85537965691787464342786601359, −3.61988559855993516146310733181, −3.29577556059048382929893190054, −3.08169825140157038054223834298, −2.36450367944707923194759578220, −1.96514617477119533849712532290, −1.39741397114043631269220843069, −0.991765080308852350426570996862, 0, 0,
0.991765080308852350426570996862, 1.39741397114043631269220843069, 1.96514617477119533849712532290, 2.36450367944707923194759578220, 3.08169825140157038054223834298, 3.29577556059048382929893190054, 3.61988559855993516146310733181, 3.85537965691787464342786601359, 4.45959312612735554465634134030, 4.57526632988669604170397183389, 5.20940538295844131186310197427, 5.30764316957079062099129793117, 6.03969934927840819380248966825, 6.21961063699938830874031650557, 6.52554882937912271436068041833, 6.82340138894936012718370614380, 7.17127843081953679290490101491, 7.32596981093123567795109009990
