| L(s) = 1 | + 6·3-s − 82·5-s − 98·7-s − 114·9-s − 340·11-s − 910·13-s − 492·15-s + 3.21e3·17-s + 674·19-s − 588·21-s − 1.10e3·23-s + 1.89e3·25-s − 126·27-s − 8.06e3·29-s − 6.21e3·31-s − 2.04e3·33-s + 8.03e3·35-s + 8.51e3·37-s − 5.46e3·39-s − 1.30e3·41-s + 1.00e4·43-s + 9.34e3·45-s − 1.27e4·47-s + 7.20e3·49-s + 1.92e4·51-s + 1.12e4·53-s + 2.78e4·55-s + ⋯ |
| L(s) = 1 | + 0.384·3-s − 1.46·5-s − 0.755·7-s − 0.469·9-s − 0.847·11-s − 1.49·13-s − 0.564·15-s + 2.69·17-s + 0.428·19-s − 0.290·21-s − 0.435·23-s + 0.607·25-s − 0.0332·27-s − 1.78·29-s − 1.16·31-s − 0.326·33-s + 1.10·35-s + 1.02·37-s − 0.574·39-s − 0.121·41-s + 0.825·43-s + 0.688·45-s − 0.841·47-s + 3/7·49-s + 1.03·51-s + 0.548·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200704 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 p T + 50 p T^{2} - 2 p^{6} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 82 T + 4826 T^{2} + 82 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 340 T + 338582 T^{2} + 340 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 70 p T + 2514 p^{2} T^{2} + 70 p^{6} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3216 T + 5412958 T^{2} - 3216 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 674 T + 4367142 T^{2} - 674 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 48 p T + 495170 p T^{2} + 48 p^{6} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 8064 T + 52795702 T^{2} + 8064 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6212 T + 51691038 T^{2} + 6212 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8512 T + 104326950 T^{2} - 8512 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 1304 T + 73546526 T^{2} + 1304 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10004 T + 99339510 T^{2} - 10004 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12748 T + 323438270 T^{2} + 12748 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11220 T + 664373806 T^{2} - 11220 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12018 T - 285266426 T^{2} - 12018 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 102738 T + 4326026138 T^{2} + 102738 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 24136 T + 1542084918 T^{2} + 24136 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 89720 T + 4576356302 T^{2} - 89720 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 55588 T + 3902743302 T^{2} + 55588 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 48824 T + 5430110622 T^{2} - 48824 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 35782 T + 4098945062 T^{2} + 35782 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 18300 T + 3539716918 T^{2} + 18300 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 69984 T + 18325482398 T^{2} + 69984 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−10.06624456124364571585692904702, −9.738355141199675348695126356487, −9.075773747329148115527048617203, −8.834561467207392660714172019460, −7.83289870836369999689621333451, −7.74713549214742054227097964158, −7.52527451886492178208321648907, −7.27670227410063270139571909460, −6.11379255168038803543413897974, −5.90170710373712533515552533410, −5.15556994131562573976959557859, −4.91259117863483593148531647481, −3.90740970557874443559069667931, −3.65046086517572875495880511182, −3.07294484959889396042394298727, −2.71924234906336862898323758395, −1.85283824060961844740658856721, −0.895458825589065300580725974104, 0, 0,
0.895458825589065300580725974104, 1.85283824060961844740658856721, 2.71924234906336862898323758395, 3.07294484959889396042394298727, 3.65046086517572875495880511182, 3.90740970557874443559069667931, 4.91259117863483593148531647481, 5.15556994131562573976959557859, 5.90170710373712533515552533410, 6.11379255168038803543413897974, 7.27670227410063270139571909460, 7.52527451886492178208321648907, 7.74713549214742054227097964158, 7.83289870836369999689621333451, 8.834561467207392660714172019460, 9.075773747329148115527048617203, 9.738355141199675348695126356487, 10.06624456124364571585692904702
