| L(s) = 1 | + 8·3-s − 36·7-s + 18·9-s − 22·11-s + 24·13-s + 8·17-s + 16·19-s − 288·21-s + 312·23-s − 8·27-s − 284·29-s − 432·31-s − 176·33-s − 12·37-s + 192·39-s − 164·41-s − 44·43-s − 152·47-s + 382·49-s + 64·51-s − 124·53-s + 128·57-s − 1.25e3·59-s + 788·61-s − 648·63-s + 752·67-s + 2.49e3·69-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 1.94·7-s + 2/3·9-s − 0.603·11-s + 0.512·13-s + 0.114·17-s + 0.193·19-s − 2.99·21-s + 2.82·23-s − 0.0570·27-s − 1.81·29-s − 2.50·31-s − 0.928·33-s − 0.0533·37-s + 0.788·39-s − 0.624·41-s − 0.156·43-s − 0.471·47-s + 1.11·49-s + 0.175·51-s − 0.321·53-s + 0.297·57-s − 2.77·59-s + 1.65·61-s − 1.29·63-s + 1.37·67-s + 4.35·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 36 T + 914 T^{2} + 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 24 T + 3938 T^{2} - 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 8 T - 7654 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 16 T + 10326 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 312 T + 48646 T^{2} - 312 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 284 T + 52718 T^{2} + 284 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 432 T + 104702 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 73598 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 p T + 130742 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 44 T + 158634 T^{2} + 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 152 T + 160406 T^{2} + 152 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 124 T + 107198 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 1256 T + 745142 T^{2} + 1256 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 788 T + 595374 T^{2} - 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 752 T + 730206 T^{2} - 752 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1312 T + 1035182 T^{2} + 1312 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1480 T + 1322730 T^{2} + 1480 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 40 T + 488814 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1068 T + 1428634 T^{2} + 1068 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 132 T - 745706 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 444 T + 1874534 T^{2} + 444 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.236852287269372570298592556047, −8.868255018603558898881681780505, −8.697990505102643742249891072183, −8.128367944863051323297909071199, −7.50438161268950006861414663283, −7.28029036160500912654170237694, −6.91437767365896136874890744025, −6.47058956638088605281649281077, −5.76813618761712496432761235672, −5.53756718138606076680279323398, −4.99495174708360247571941581753, −4.25179585954205962223125668975, −3.53343015992190064628514531595, −3.40691031488742424185551637381, −2.91430133487395756053670041068, −2.76350554130163053658792587797, −1.82086466567617497167348347900, −1.29005186019088573770596383251, 0, 0,
1.29005186019088573770596383251, 1.82086466567617497167348347900, 2.76350554130163053658792587797, 2.91430133487395756053670041068, 3.40691031488742424185551637381, 3.53343015992190064628514531595, 4.25179585954205962223125668975, 4.99495174708360247571941581753, 5.53756718138606076680279323398, 5.76813618761712496432761235672, 6.47058956638088605281649281077, 6.91437767365896136874890744025, 7.28029036160500912654170237694, 7.50438161268950006861414663283, 8.128367944863051323297909071199, 8.697990505102643742249891072183, 8.868255018603558898881681780505, 9.236852287269372570298592556047
