L(s) = 1 | + 3·2-s + 5·4-s + 6·8-s + 9-s + 6·11-s + 4·13-s + 4·16-s − 6·17-s + 3·18-s + 6·19-s + 18·22-s + 12·23-s + 5·25-s + 12·26-s + 10·31-s − 18·34-s + 5·36-s + 4·37-s + 18·38-s − 32·43-s + 30·44-s + 36·46-s − 6·47-s + 15·50-s + 20·52-s + 6·53-s + 6·59-s + ⋯ |
L(s) = 1 | + 2.12·2-s + 5/2·4-s + 2.12·8-s + 1/3·9-s + 1.80·11-s + 1.10·13-s + 16-s − 1.45·17-s + 0.707·18-s + 1.37·19-s + 3.83·22-s + 2.50·23-s + 25-s + 2.35·26-s + 1.79·31-s − 3.08·34-s + 5/6·36-s + 0.657·37-s + 2.91·38-s − 4.87·43-s + 4.52·44-s + 5.30·46-s − 0.875·47-s + 2.12·50-s + 2.77·52-s + 0.824·53-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(15.69775001\) |
\(L(\frac12)\) |
\(\approx\) |
\(15.69775001\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 - T^{2} + p^{2} T^{4} ) \) |
| 3 | $C_2^3$ | \( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 13 T^{2} - 66 T^{3} - 372 T^{4} - 66 p T^{5} + 13 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 3 T - 10 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 67 T^{2} - 372 T^{3} + 2088 T^{4} - 372 p T^{5} + 67 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T - 17 T^{2} + 164 T^{3} - 872 T^{4} + 164 p T^{5} - 17 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 6 T - p T^{2} - 66 T^{3} + 2988 T^{4} - 66 p T^{5} - p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 59 T^{2} + 66 T^{3} + 4308 T^{4} + 66 p T^{5} - 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T - 71 T^{2} + 66 T^{3} + 5844 T^{4} + 66 p T^{5} - 71 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T - 53 T^{2} + 232 T^{3} + 3688 T^{4} + 232 p T^{5} - 53 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T - 65 T^{2} - 232 T^{3} + 5344 T^{4} - 232 p T^{5} - 65 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 89 | $C_2^3$ | \( 1 - 173 T^{2} + 22008 T^{4} - 173 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.29659759714342890931595266705, −7.07142712545683308407459715000, −6.93076587463328469304648565271, −6.91975938625982184569873604364, −6.63557916723416195666476517112, −6.43656087010451697196656727330, −6.11586339675428518223589975536, −6.06978223505032457584478250949, −5.71177127466275601571271082886, −5.09806230723001197364171513873, −5.07530185526766253034894590276, −4.90336449348399530239699003410, −4.84654172195487196766523018450, −4.45700126473678639780881347558, −4.07349417103731306166888072790, −3.95607249836668106297901538249, −3.58190473073688495419791505823, −3.22908109928343167093342924606, −3.20905148485737465340353365424, −2.91831266741466623778743797806, −2.45378857554826320941896047546, −1.97072174160825553029468564769, −1.65101066267164557495071182343, −1.07959789309069802873356311898, −0.949291099823486747223309816262,
0.949291099823486747223309816262, 1.07959789309069802873356311898, 1.65101066267164557495071182343, 1.97072174160825553029468564769, 2.45378857554826320941896047546, 2.91831266741466623778743797806, 3.20905148485737465340353365424, 3.22908109928343167093342924606, 3.58190473073688495419791505823, 3.95607249836668106297901538249, 4.07349417103731306166888072790, 4.45700126473678639780881347558, 4.84654172195487196766523018450, 4.90336449348399530239699003410, 5.07530185526766253034894590276, 5.09806230723001197364171513873, 5.71177127466275601571271082886, 6.06978223505032457584478250949, 6.11586339675428518223589975536, 6.43656087010451697196656727330, 6.63557916723416195666476517112, 6.91975938625982184569873604364, 6.93076587463328469304648565271, 7.07142712545683308407459715000, 7.29659759714342890931595266705