| L(s) = 1 | + 3·2-s + 5·4-s − 8·7-s + 6·8-s + 9-s + 6·11-s − 4·13-s − 24·14-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 − 40·28-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 + 34·49-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 5/2·4-s − 3.02·7-s + 2.12·8-s + 1/3·9-s + 1.80·11-s − 1.10·13-s − 6.41·14-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 − 7.55·28-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 + 34/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.126927175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126927175\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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
−10.41507271308888056682771179061, −10.19511879307132675294626325601, −9.963985434852701662601982436292, −9.576703550189450260791544118253, −9.365255247777837370764012818202, −9.025745998523202722389118705428, −8.946273652550477929618178825074, −8.458333016419311493913696830101, −7.974674709273874811639719904103, −7.28707580737416211507804133607, −7.00073560932273874586189697192, −6.98263698841547473536537244743, −6.77337655772861857255230814779, −6.37522657177174766914774324039, −5.96738277904419845478407978227, −5.94049446538056836979371913842, −5.09779491066717041247569614310, −5.05436121462349775984850677814, −4.64406321105409139521068003200, −3.97646047461772695807843743915, −3.77734419038093641113834944118, −3.35327604878179764201177913968, −3.00255312100191105218792907301, −2.82081953332938905818375982573, −1.69057622283079173193073362078,
1.69057622283079173193073362078, 2.82081953332938905818375982573, 3.00255312100191105218792907301, 3.35327604878179764201177913968, 3.77734419038093641113834944118, 3.97646047461772695807843743915, 4.64406321105409139521068003200, 5.05436121462349775984850677814, 5.09779491066717041247569614310, 5.94049446538056836979371913842, 5.96738277904419845478407978227, 6.37522657177174766914774324039, 6.77337655772861857255230814779, 6.98263698841547473536537244743, 7.00073560932273874586189697192, 7.28707580737416211507804133607, 7.974674709273874811639719904103, 8.458333016419311493913696830101, 8.946273652550477929618178825074, 9.025745998523202722389118705428, 9.365255247777837370764012818202, 9.576703550189450260791544118253, 9.963985434852701662601982436292, 10.19511879307132675294626325601, 10.41507271308888056682771179061
