| L(s) = 1 | + 10·3-s + 10·5-s − 16·7-s + 39·9-s + 26·11-s + 26·13-s + 100·15-s + 60·17-s + 220·19-s − 160·21-s + 52·23-s − 143·25-s + 50·27-s − 58·29-s − 294·31-s + 260·33-s − 160·35-s − 312·37-s + 260·39-s + 40·41-s + 322·43-s + 390·45-s − 130·47-s − 294·49-s + 600·51-s − 1.00e3·53-s + 260·55-s + ⋯ |
| L(s) = 1 | + 1.92·3-s + 0.894·5-s − 0.863·7-s + 13/9·9-s + 0.712·11-s + 0.554·13-s + 1.72·15-s + 0.856·17-s + 2.65·19-s − 1.66·21-s + 0.471·23-s − 1.14·25-s + 0.356·27-s − 0.371·29-s − 1.70·31-s + 1.37·33-s − 0.772·35-s − 1.38·37-s + 1.06·39-s + 0.152·41-s + 1.14·43-s + 1.29·45-s − 0.403·47-s − 6/7·49-s + 1.64·51-s − 2.59·53-s + 0.637·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3444736 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.918902753\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.918902753\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 10 T + 61 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 p T + 243 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 16 T + 550 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 26 T + 93 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 p T + 19 p^{2} T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 60 T + 10078 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 220 T + 23770 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 52 T + 20402 T^{2} - 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 294 T + 2363 p T^{2} + 294 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 312 T + 119370 T^{2} + 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 40 T + 100154 T^{2} - 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 322 T + 126453 T^{2} - 322 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 130 T + 126173 T^{2} + 130 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1002 T + 518987 T^{2} + 1002 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 900 T + 490250 T^{2} - 900 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 948 T + 615270 T^{2} - 948 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 320 T + 158614 T^{2} + 320 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 660 T + 822410 T^{2} + 660 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 648 T + 63810 T^{2} - 648 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 258 T + 770157 T^{2} - 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1212 T + 1502618 T^{2} + 1212 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 760 T + 1009370 T^{2} - 760 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 24 T + 1157322 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.111354167251516207551521307130, −8.955990410472242526800123499670, −8.198121888834651942471490852510, −8.074222267481988160080804991160, −7.49914669265196095635401410743, −7.31687091415235579985720472760, −6.77436844831257880317041922202, −6.39841819151837979958539482796, −5.72054936773169372076618004715, −5.53498564000533182793210320369, −5.23181109589638870392600909325, −4.40695474363111085089052042324, −3.61672943294744400938941266858, −3.58856983184278774633463402859, −3.19597087224640337507455113700, −2.88202057417799768649202659400, −2.10526997096421772415782900806, −1.75395916565761060623624003151, −1.25614658876055763001344382665, −0.52912452322119245020990641007,
0.52912452322119245020990641007, 1.25614658876055763001344382665, 1.75395916565761060623624003151, 2.10526997096421772415782900806, 2.88202057417799768649202659400, 3.19597087224640337507455113700, 3.58856983184278774633463402859, 3.61672943294744400938941266858, 4.40695474363111085089052042324, 5.23181109589638870392600909325, 5.53498564000533182793210320369, 5.72054936773169372076618004715, 6.39841819151837979958539482796, 6.77436844831257880317041922202, 7.31687091415235579985720472760, 7.49914669265196095635401410743, 8.074222267481988160080804991160, 8.198121888834651942471490852510, 8.955990410472242526800123499670, 9.111354167251516207551521307130
