| L(s) = 1 | + 5-s + 3·7-s + 3·11-s + 12·13-s − 17-s − 2·19-s + 8·23-s + 25-s − 4·29-s − 6·31-s + 3·35-s + 6·37-s − 6·41-s − 43-s + 5·47-s + 3·49-s + 12·53-s + 3·55-s − 8·59-s + 9·61-s + 12·65-s + 24·67-s + 7·73-s + 9·77-s − 4·79-s + 4·83-s − 85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s + 0.904·11-s + 3.32·13-s − 0.242·17-s − 0.458·19-s + 1.66·23-s + 1/5·25-s − 0.742·29-s − 1.07·31-s + 0.507·35-s + 0.986·37-s − 0.937·41-s − 0.152·43-s + 0.729·47-s + 3/7·49-s + 1.64·53-s + 0.404·55-s − 1.04·59-s + 1.15·61-s + 1.48·65-s + 2.93·67-s + 0.819·73-s + 1.02·77-s − 0.450·79-s + 0.439·83-s − 0.108·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.480399493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.480399493\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - T - p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 6 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + T + 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 42 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 90 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 9 T + 132 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 148 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.921911727116791371426556241022, −8.631340419532668056649987861946, −8.313422434577610782703747839098, −8.163241199022309603922342786487, −7.43932233232358196747629989360, −7.06630649185132370527665321635, −6.68932255664833789415208962646, −6.33902792531288865110626768653, −5.92766480130418452316482589468, −5.63465522142915149935223728988, −5.11659551098610445376251249123, −4.82933858274108458243299412201, −4.03433540767028760332384776868, −3.83944256472822148577496343454, −3.61821290788363792929502287498, −2.93772816651841095731164005897, −2.15565656473715690413216233405, −1.78932366113614628914885984326, −1.04398047849916943650448216363, −1.02783915716178684131355557050,
1.02783915716178684131355557050, 1.04398047849916943650448216363, 1.78932366113614628914885984326, 2.15565656473715690413216233405, 2.93772816651841095731164005897, 3.61821290788363792929502287498, 3.83944256472822148577496343454, 4.03433540767028760332384776868, 4.82933858274108458243299412201, 5.11659551098610445376251249123, 5.63465522142915149935223728988, 5.92766480130418452316482589468, 6.33902792531288865110626768653, 6.68932255664833789415208962646, 7.06630649185132370527665321635, 7.43932233232358196747629989360, 8.163241199022309603922342786487, 8.313422434577610782703747839098, 8.631340419532668056649987861946, 8.921911727116791371426556241022
