| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 5-s + 2·6-s + 3·7-s + 4·8-s + 2·10-s − 5·11-s + 3·12-s − 6·13-s + 6·14-s + 15-s + 5·16-s − 8·17-s + 4·19-s + 3·20-s + 3·21-s − 10·22-s + 16·23-s + 4·24-s − 12·26-s − 27-s + 9·28-s − 2·29-s + 2·30-s + 7·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.447·5-s + 0.816·6-s + 1.13·7-s + 1.41·8-s + 0.632·10-s − 1.50·11-s + 0.866·12-s − 1.66·13-s + 1.60·14-s + 0.258·15-s + 5/4·16-s − 1.94·17-s + 0.917·19-s + 0.670·20-s + 0.654·21-s − 2.13·22-s + 3.33·23-s + 0.816·24-s − 2.35·26-s − 0.192·27-s + 1.70·28-s − 0.371·29-s + 0.365·30-s + 1.25·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.197709179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.197709179\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 31 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 5 T + 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 5 T - 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 3 T - 74 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−10.30478829125647750313917427193, −10.12007160013701459727914195549, −9.347403971485160717193803877670, −8.844716599488246756391889802239, −8.824277912017998811290746832557, −7.979285671058630070901909676871, −7.50807883207395849046759655625, −7.36984083209812424480092532595, −6.93026350648617093171363735046, −6.44200231245006661185708823773, −5.52674926280873650344152585339, −5.46959263109360948546350358763, −4.89841119727973533320497935899, −4.61181462844985176584149588873, −4.32394364756848824847662722274, −3.30873572242856530579676389920, −2.63069543971932632781557599185, −2.62841328024383300917228078445, −2.08174974302417422411168557278, −0.996437145237133674360229016805,
0.996437145237133674360229016805, 2.08174974302417422411168557278, 2.62841328024383300917228078445, 2.63069543971932632781557599185, 3.30873572242856530579676389920, 4.32394364756848824847662722274, 4.61181462844985176584149588873, 4.89841119727973533320497935899, 5.46959263109360948546350358763, 5.52674926280873650344152585339, 6.44200231245006661185708823773, 6.93026350648617093171363735046, 7.36984083209812424480092532595, 7.50807883207395849046759655625, 7.979285671058630070901909676871, 8.824277912017998811290746832557, 8.844716599488246756391889802239, 9.347403971485160717193803877670, 10.12007160013701459727914195549, 10.30478829125647750313917427193
