| L(s) = 1 | − 2·2-s − 2·3-s − 4-s − 14·5-s + 4·6-s + 16·7-s + 16·9-s + 28·10-s + 4·11-s + 2·12-s − 26·13-s − 32·14-s + 28·15-s + 12·16-s − 12·17-s − 32·18-s + 10·19-s + 14·20-s − 32·21-s − 8·22-s + 18·23-s + 98·25-s + 52·26-s − 52·27-s − 16·28-s + 2·29-s − 56·30-s + ⋯ |
| L(s) = 1 | − 2-s − 2/3·3-s − 1/4·4-s − 2.79·5-s + 2/3·6-s + 16/7·7-s + 16/9·9-s + 14/5·10-s + 4/11·11-s + 1/6·12-s − 2·13-s − 2.28·14-s + 1.86·15-s + 3/4·16-s − 0.705·17-s − 1.77·18-s + 0.526·19-s + 7/10·20-s − 1.52·21-s − 0.363·22-s + 0.782·23-s + 3.91·25-s + 2·26-s − 1.92·27-s − 4/7·28-s + 2/29·29-s − 1.86·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28561 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1427101370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1427101370\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p T + 5 T^{2} + 3 p^{2} T^{3} + 17 T^{4} + 3 p^{4} T^{5} + 5 p^{4} T^{6} + p^{7} T^{7} + p^{8} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 + 2 T - 4 p T^{2} - 4 T^{3} + 139 T^{4} - 4 p^{2} T^{5} - 4 p^{5} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 6 T + 11 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} )( 1 + 8 T + 39 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 16 T + 164 T^{2} - 1236 T^{3} + 8927 T^{4} - 1236 p^{2} T^{5} + 164 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 200 T^{2} + 960 T^{3} + 17471 T^{4} + 960 p^{2} T^{5} + 200 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 413 T^{2} + 4380 T^{3} + 63576 T^{4} + 4380 p^{2} T^{5} + 413 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 10 T + 74 T^{2} - 1752 T^{3} - 96625 T^{4} - 1752 p^{2} T^{5} + 74 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 18 T + 968 T^{2} - 15480 T^{3} + 516891 T^{4} - 15480 p^{2} T^{5} + 968 p^{4} T^{6} - 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 1571 T^{2} + 214 T^{3} + 1769980 T^{4} + 214 p^{2} T^{5} - 1571 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 18300 T^{3} + 1672334 T^{4} + 18300 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 68 T + 41 p T^{2} - 93168 T^{3} - 5925028 T^{4} - 93168 p^{2} T^{5} + 41 p^{5} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 100 T + 3461 T^{2} + 7272 T^{3} - 4096804 T^{4} + 7272 p^{2} T^{5} + 3461 p^{4} T^{6} - 100 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 90 T + 4549 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 68 T + 2312 T^{2} + 148716 T^{3} + 9565454 T^{4} + 148716 p^{2} T^{5} + 2312 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 64 T + 4455 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 164 T + 6980 T^{2} - 551844 T^{3} - 68910913 T^{4} - 551844 p^{2} T^{5} + 6980 p^{4} T^{6} + 164 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 124 T + 5413 T^{2} + 312604 T^{3} + 28201432 T^{4} + 312604 p^{2} T^{5} + 5413 p^{4} T^{6} + 124 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 118 T + 10706 T^{2} - 763488 T^{3} + 51177839 T^{4} - 763488 p^{2} T^{5} + 10706 p^{4} T^{6} - 118 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 86 T + 4658 T^{2} + 258336 T^{3} + 1457087 T^{4} + 258336 p^{2} T^{5} + 4658 p^{4} T^{6} + 86 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 58 T + 1682 T^{2} - 201144 T^{3} + 20590727 T^{4} - 201144 p^{2} T^{5} + 1682 p^{4} T^{6} - 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 20 T + 7290 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 188 T + 17672 T^{2} + 1935084 T^{3} + 200304482 T^{4} + 1935084 p^{2} T^{5} + 17672 p^{4} T^{6} + 188 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 110 T + 12050 T^{2} + 1194180 T^{3} + 87746159 T^{4} + 1194180 p^{2} T^{5} + 12050 p^{4} T^{6} + 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 178 T + 13250 T^{2} - 318828 T^{3} - 39375793 T^{4} - 318828 p^{2} T^{5} + 13250 p^{4} T^{6} - 178 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.29071013712631202961537634294, −14.92450749565529743467736844708, −14.60793667814246444314596565503, −14.23948615269701998674708152208, −14.02871999898276824260912970034, −12.87656638065065882475295154215, −12.73506263469685486997756218993, −12.35508748671146947038322297182, −11.94096394180814851765994738485, −11.79909410586806601335509550978, −11.17170965216970607950473066331, −10.93174628761151014090250123475, −10.80114878676248533836218564486, −9.949203577202307198335670126866, −9.441057435701829335531531376602, −8.882317382453934682412255272055, −8.748529320555201240166277877670, −7.82745848784865698386406467506, −7.51474711131236354201642582585, −7.49426845231199612039305488387, −7.14798786949140281813477201047, −5.69553781794938833798720680035, −4.65942459653037270782916514650, −4.61993023668330184227022435093, −3.95683416420677932391690554448,
3.95683416420677932391690554448, 4.61993023668330184227022435093, 4.65942459653037270782916514650, 5.69553781794938833798720680035, 7.14798786949140281813477201047, 7.49426845231199612039305488387, 7.51474711131236354201642582585, 7.82745848784865698386406467506, 8.748529320555201240166277877670, 8.882317382453934682412255272055, 9.441057435701829335531531376602, 9.949203577202307198335670126866, 10.80114878676248533836218564486, 10.93174628761151014090250123475, 11.17170965216970607950473066331, 11.79909410586806601335509550978, 11.94096394180814851765994738485, 12.35508748671146947038322297182, 12.73506263469685486997756218993, 12.87656638065065882475295154215, 14.02871999898276824260912970034, 14.23948615269701998674708152208, 14.60793667814246444314596565503, 14.92450749565529743467736844708, 15.29071013712631202961537634294
