| L(s) = 1 | + 6·2-s − 3·3-s + 17·4-s + 3·5-s − 18·6-s + 30·8-s + 7·9-s + 18·10-s + 3·11-s − 51·12-s + 10·13-s − 9·15-s + 40·16-s − 12·17-s + 42·18-s + 3·19-s + 51·20-s + 18·22-s − 90·24-s + 11·25-s + 60·26-s − 18·27-s − 3·29-s − 54·30-s + 4·31-s + 54·32-s − 9·33-s + ⋯ |
| L(s) = 1 | + 4.24·2-s − 1.73·3-s + 17/2·4-s + 1.34·5-s − 7.34·6-s + 10.6·8-s + 7/3·9-s + 5.69·10-s + 0.904·11-s − 14.7·12-s + 2.77·13-s − 2.32·15-s + 10·16-s − 2.91·17-s + 9.89·18-s + 0.688·19-s + 11.4·20-s + 3.83·22-s − 18.3·24-s + 11/5·25-s + 11.7·26-s − 3.46·27-s − 0.557·29-s − 9.85·30-s + 0.718·31-s + 9.54·32-s − 1.56·33-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
| \(L(1)\) |
\(\approx\) |
\(27.40099246\) |
| \(L(\frac12)\) |
\(\approx\) |
\(27.40099246\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$ | \( ( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 3 | $D_4\times C_2$ | \( 1 + p T + 2 T^{2} + p T^{3} + 13 T^{4} + p^{2} T^{5} + 2 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 3 T - 2 T^{2} - 3 T^{3} + 51 T^{4} - 3 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 3 T - 4 T^{2} + 27 T^{3} - 51 T^{4} + 27 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 3 T - 20 T^{2} + 27 T^{3} + 309 T^{4} + 27 p T^{5} - 20 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T - 20 T^{2} - 3 p T^{3} - 9 p T^{4} - 3 p^{2} T^{5} - 20 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T - 5 T^{2} + 164 T^{3} - 1016 T^{4} + 164 p T^{5} - 5 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6 T - 50 T^{2} - 24 T^{3} + 4239 T^{4} - 24 p T^{5} - 50 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T + 34 T^{2} - 475 T^{3} - 2843 T^{4} - 475 p T^{5} + 34 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 89 T^{2} + 5712 T^{4} - 89 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 7 T^{2} + 372 T^{3} + 8328 T^{4} + 372 p T^{5} + 7 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 113 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 19 T^{2} + 108 T^{3} + 1488 T^{4} + 108 p T^{5} + 19 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - 6 T + 10 T^{2} + 696 T^{3} - 6921 T^{4} + 696 p T^{5} + 10 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 17 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 21 T + 257 T^{2} + 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 31 T + 538 T^{2} - 7099 T^{3} + 76303 T^{4} - 7099 p T^{5} + 538 p^{2} T^{6} - 31 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.14433896343688245098706975224, −7.01347246916965416940792969145, −6.73450760598949925423430825685, −6.66703667723165291746282529119, −6.35814881585428269160385938384, −6.07844255161305555182287926059, −6.04999865541552475322390782921, −5.82341529911665745838019732906, −5.69975358444092434768934146623, −5.54502247221733473876116968591, −5.02405026437754056215126054913, −4.93219282137661721455739809394, −4.59281074543405768618620541542, −4.45663315585409413447069379874, −4.24981985766570571159745757928, −4.21032971408335870801932264405, −3.75982478259556372361563691050, −3.60465091763564915509551465884, −3.24735040076625536118415992436, −2.76883349772188162769633074744, −2.72642307923044852406261197561, −1.96810569160220177028483107403, −1.66340222706922530234289163376, −1.32589893783868442776978775975, −0.867071119678678533980955765048,
0.867071119678678533980955765048, 1.32589893783868442776978775975, 1.66340222706922530234289163376, 1.96810569160220177028483107403, 2.72642307923044852406261197561, 2.76883349772188162769633074744, 3.24735040076625536118415992436, 3.60465091763564915509551465884, 3.75982478259556372361563691050, 4.21032971408335870801932264405, 4.24981985766570571159745757928, 4.45663315585409413447069379874, 4.59281074543405768618620541542, 4.93219282137661721455739809394, 5.02405026437754056215126054913, 5.54502247221733473876116968591, 5.69975358444092434768934146623, 5.82341529911665745838019732906, 6.04999865541552475322390782921, 6.07844255161305555182287926059, 6.35814881585428269160385938384, 6.66703667723165291746282529119, 6.73450760598949925423430825685, 7.01347246916965416940792969145, 7.14433896343688245098706975224
