| L(s) = 1 | − 4·2-s − 2·3-s + 11·4-s + 14·5-s + 8·6-s + 20·7-s − 18·8-s + 16·9-s − 56·10-s − 28·11-s − 22·12-s − 80·14-s − 28·15-s + 24·16-s + 12·17-s − 64·18-s + 14·19-s + 154·20-s − 40·21-s + 112·22-s − 18·23-s + 36·24-s + 98·25-s − 52·27-s + 220·28-s + 2·29-s + 112·30-s + ⋯ |
| L(s) = 1 | − 2·2-s − 2/3·3-s + 11/4·4-s + 14/5·5-s + 4/3·6-s + 20/7·7-s − 9/4·8-s + 16/9·9-s − 5.59·10-s − 2.54·11-s − 1.83·12-s − 5.71·14-s − 1.86·15-s + 3/2·16-s + 0.705·17-s − 3.55·18-s + 0.736·19-s + 7.69·20-s − 1.90·21-s + 5.09·22-s − 0.782·23-s + 3/2·24-s + 3.91·25-s − 1.92·27-s + 55/7·28-s + 2/29·29-s + 3.73·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.877515579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.877515579\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + 5 T^{2} - 3 p T^{3} - 31 T^{4} - 3 p^{3} T^{5} + 5 p^{4} T^{6} + p^{8} T^{7} + p^{8} T^{8} \) |
| 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$ | \( ( 1 - 8 T + 39 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )( 1 - 6 T + 11 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} ) \) |
| 7 | $D_4\times C_2$ | \( 1 - 20 T + 164 T^{2} - 636 T^{3} + 1871 T^{4} - 636 p^{2} T^{5} + 164 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 28 T + 200 T^{2} - 3456 T^{3} - 75457 T^{4} - 3456 p^{2} T^{5} + 200 p^{4} T^{6} + 28 p^{6} T^{7} + p^{8} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 - 14 T + 74 T^{2} + 1224 T^{3} - 131281 T^{4} + 1224 p^{2} T^{5} + 74 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 - 38 T + 41 p T^{2} + 52158 T^{3} - 1571608 T^{4} + 52158 p^{2} T^{5} + 41 p^{5} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 62 T + 3461 T^{2} - 155178 T^{3} + 6251432 T^{4} - 155178 p^{2} T^{5} + 3461 p^{4} T^{6} - 62 p^{6} T^{7} + p^{8} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 - 32 T + 6980 T^{2} + 13812 T^{3} + 21149519 T^{4} + 13812 p^{2} T^{5} + 6980 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 - 170 T + 10706 T^{2} - 134496 T^{3} - 16049425 T^{4} - 134496 p^{2} T^{5} + 10706 p^{4} T^{6} - 170 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 106 T + 4658 T^{2} + 26976 T^{3} - 17900353 T^{4} + 26976 p^{2} T^{5} + 4658 p^{4} T^{6} + 106 p^{6} T^{7} + p^{8} T^{8} \) |
| 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 | $D_4\times C_2$ | \( 1 + 190 T + 12050 T^{2} - 309180 T^{3} - 102357841 T^{4} - 309180 p^{2} T^{5} + 12050 p^{4} T^{6} + 190 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 146 T + 13250 T^{2} - 1056876 T^{3} + 58176719 T^{4} - 1056876 p^{2} T^{5} + 13250 p^{4} T^{6} - 146 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−9.040011637123384823407152240104, −9.000148638690325253654735814744, −8.501090402437619493381970771408, −8.317949786581961005669130165963, −7.898256606451876277208848735666, −7.889818464229087298376647178301, −7.60374207641007084727766993874, −7.32200926051012133327265393599, −7.09369308750657129687895616624, −6.78854021486654896461798063786, −6.34528316004958644232981136107, −5.78871200579765962118996911515, −5.65095994535099092591995492266, −5.49169322984298131418301772891, −5.33333736634378040900743832411, −4.74770530015728468033202865946, −4.67005068698950186276776773519, −4.24479820132136636998290404573, −3.42028432513463735293988729918, −2.65880287904470937816067137247, −2.27605262298047352401303574604, −1.97317784964353103198762571803, −1.75806075413476988627121443342, −1.30769029412786992155024163499, −0.896128000723338810231908931665,
0.896128000723338810231908931665, 1.30769029412786992155024163499, 1.75806075413476988627121443342, 1.97317784964353103198762571803, 2.27605262298047352401303574604, 2.65880287904470937816067137247, 3.42028432513463735293988729918, 4.24479820132136636998290404573, 4.67005068698950186276776773519, 4.74770530015728468033202865946, 5.33333736634378040900743832411, 5.49169322984298131418301772891, 5.65095994535099092591995492266, 5.78871200579765962118996911515, 6.34528316004958644232981136107, 6.78854021486654896461798063786, 7.09369308750657129687895616624, 7.32200926051012133327265393599, 7.60374207641007084727766993874, 7.889818464229087298376647178301, 7.898256606451876277208848735666, 8.317949786581961005669130165963, 8.501090402437619493381970771408, 9.000148638690325253654735814744, 9.040011637123384823407152240104
