| L(s) = 1 | + 3-s + 3·4-s + 3·5-s + 9-s + 9·11-s + 3·12-s − 14·13-s + 3·15-s + 4·16-s − 12·17-s − 9·19-s + 9·20-s + 12·23-s − 3·25-s − 4·27-s + 9·29-s + 30·31-s + 9·33-s + 3·36-s − 14·39-s − 18·41-s − 5·43-s + 27·44-s + 3·45-s + 24·47-s + 4·48-s − 12·51-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3/2·4-s + 1.34·5-s + 1/3·9-s + 2.71·11-s + 0.866·12-s − 3.88·13-s + 0.774·15-s + 16-s − 2.91·17-s − 2.06·19-s + 2.01·20-s + 2.50·23-s − 3/5·25-s − 0.769·27-s + 1.67·29-s + 5.38·31-s + 1.56·33-s + 1/2·36-s − 2.24·39-s − 2.81·41-s − 0.762·43-s + 4.07·44-s + 0.447·45-s + 3.50·47-s + 0.577·48-s − 1.68·51-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
| \(L(1)\) |
\(\approx\) |
\(6.307219756\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.307219756\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} ) \) |
| 3 | $D_4\times C_2$ | \( 1 - T + 5 T^{3} - 11 T^{4} + 5 p T^{5} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 3 T + 12 T^{2} - 27 T^{3} + 71 T^{4} - 27 p T^{5} + 12 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 9 T + 54 T^{2} - 243 T^{3} + 905 T^{4} - 243 p T^{5} + 54 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 19 | $D_4\times C_2$ | \( 1 + 9 T + 56 T^{2} + 261 T^{3} + 993 T^{4} + 261 p T^{5} + 56 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 9 T + 8 T^{2} - 135 T^{3} + 2139 T^{4} - 135 p T^{5} + 8 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 210 T^{2} + 1836 T^{3} + 13151 T^{4} + 1836 p T^{5} + 210 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 5 T - 20 T^{2} - 205 T^{3} - 899 T^{4} - 205 p T^{5} - 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 24 T + 327 T^{2} - 3240 T^{3} + 25040 T^{4} - 3240 p T^{5} + 327 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T + 5 T^{2} + 450 T^{3} - 3756 T^{4} + 450 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6 T^{2} + 5627 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 2 T + 70 T^{2} - 376 T^{3} + 391 T^{4} - 376 p T^{5} + 70 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + 4 T + p T^{2} )^{2}( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 150 T^{2} - 828 T^{3} + 14855 T^{4} - 828 p T^{5} + 150 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 270 T^{2} + 31667 T^{4} - 270 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 87 T^{2} + 1853 T^{4} - 87 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 39 T + 812 T^{2} - 11895 T^{3} + 132795 T^{4} - 11895 p T^{5} + 812 p^{2} T^{6} - 39 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.41797706601794476924588567162, −7.37334985932552469135494613465, −7.07829446431214611821892572457, −6.73026388398757019681449655512, −6.56139384277718308642843366832, −6.45740792489184069708560120282, −6.42883799761839721886692724428, −6.39050970461606434915508643514, −5.91958148762236628856279437785, −5.28170885232712290420499610630, −5.16368294417713953081998970209, −4.94130153665576688602015884051, −4.56559258033460906841430322472, −4.32948954678276299851584239643, −4.22720016079006004795848593384, −4.18422274463419537881382882272, −3.29720734956952129651356291656, −3.10690378896087616522524560119, −2.65797190259664534910305746339, −2.43295887203806432238362271193, −2.26865212919591692733733551697, −2.20527056451832008666077908539, −1.78838542180004913324572604439, −1.24420679570262469178396602579, −0.63068482637995740321499257054,
0.63068482637995740321499257054, 1.24420679570262469178396602579, 1.78838542180004913324572604439, 2.20527056451832008666077908539, 2.26865212919591692733733551697, 2.43295887203806432238362271193, 2.65797190259664534910305746339, 3.10690378896087616522524560119, 3.29720734956952129651356291656, 4.18422274463419537881382882272, 4.22720016079006004795848593384, 4.32948954678276299851584239643, 4.56559258033460906841430322472, 4.94130153665576688602015884051, 5.16368294417713953081998970209, 5.28170885232712290420499610630, 5.91958148762236628856279437785, 6.39050970461606434915508643514, 6.42883799761839721886692724428, 6.45740792489184069708560120282, 6.56139384277718308642843366832, 6.73026388398757019681449655512, 7.07829446431214611821892572457, 7.37334985932552469135494613465, 7.41797706601794476924588567162
