| L(s) = 1 | + 9·5-s − 5·7-s + 18·9-s + 3·11-s − 15·17-s + 38·19-s + 60·23-s + 25·25-s − 45·35-s − 85·43-s + 162·45-s + 75·47-s + 49·49-s + 27·55-s − 103·61-s − 90·63-s + 25·73-s − 15·77-s + 243·81-s − 180·83-s − 135·85-s + 342·95-s + 54·99-s − 204·101-s + 540·115-s + 75·119-s + 121·121-s + ⋯ |
| L(s) = 1 | + 9/5·5-s − 5/7·7-s + 2·9-s + 3/11·11-s − 0.882·17-s + 2·19-s + 2.60·23-s + 25-s − 9/7·35-s − 1.97·43-s + 18/5·45-s + 1.59·47-s + 49-s + 0.490·55-s − 1.68·61-s − 1.42·63-s + 0.342·73-s − 0.194·77-s + 3·81-s − 2.16·83-s − 1.58·85-s + 18/5·95-s + 6/11·99-s − 2.01·101-s + 4.69·115-s + 0.630·119-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.884209768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.884209768\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 9 T + 56 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 5 T - 24 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 112 T^{2} - 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T - 64 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 85 T + 5376 T^{2} + 85 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 75 T + 3416 T^{2} - 75 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 103 T + 6888 T^{2} + 103 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T - 4704 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 90 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.74523354909544479534557465134, −11.18050881427607056761810843574, −10.59507105070433135443924708194, −10.23621484319480102919707097708, −9.691602568656879203804252937433, −9.612821768208381558398961534541, −9.060086781445618782109225072665, −8.776457948892287430554478166779, −7.61031378594249260159626787324, −7.30229502036912534084021195732, −6.69542809881748269000581762024, −6.59511370773948215819697904413, −5.72280140819789620401898673931, −5.25281726962846126040993325112, −4.76151138666797696054809358618, −4.01277597196325535274215724036, −3.20240954051614630581050321003, −2.55403636784693369331050761465, −1.55585952996864030485385190200, −1.13979888369494711710710276437,
1.13979888369494711710710276437, 1.55585952996864030485385190200, 2.55403636784693369331050761465, 3.20240954051614630581050321003, 4.01277597196325535274215724036, 4.76151138666797696054809358618, 5.25281726962846126040993325112, 5.72280140819789620401898673931, 6.59511370773948215819697904413, 6.69542809881748269000581762024, 7.30229502036912534084021195732, 7.61031378594249260159626787324, 8.776457948892287430554478166779, 9.060086781445618782109225072665, 9.612821768208381558398961534541, 9.691602568656879203804252937433, 10.23621484319480102919707097708, 10.59507105070433135443924708194, 11.18050881427607056761810843574, 11.74523354909544479534557465134
