| L(s) = 1 | − 2·2-s − 3-s + 4-s + 4·5-s + 2·6-s − 8·10-s − 12-s − 4·15-s − 2·17-s + 3·19-s + 4·20-s + 3·23-s + 10·25-s + 8·30-s − 31-s + 2·32-s + 4·34-s − 6·38-s − 6·46-s − 2·47-s − 49-s − 20·50-s + 2·51-s − 2·53-s − 3·57-s − 4·60-s − 2·61-s + ⋯ |
| L(s) = 1 | − 2·2-s − 3-s + 4-s + 4·5-s + 2·6-s − 8·10-s − 12-s − 4·15-s − 2·17-s + 3·19-s + 4·20-s + 3·23-s + 10·25-s + 8·30-s − 31-s + 2·32-s + 4·34-s − 6·38-s − 6·46-s − 2·47-s − 49-s − 20·50-s + 2·51-s − 2·53-s − 3·57-s − 4·60-s − 2·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2121640765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2121640765\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 31 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 7 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 19 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 53 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 83 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−8.559946486275999449820491772239, −8.303034948489753777165551496494, −7.61701511355788899978332756371, −7.48309950394572100867730483641, −7.26146128510121315295082439118, −6.93056979503681191081223231883, −6.78319011778848336585267442274, −6.32357887775229766665561452861, −6.19801910734799788340793746357, −6.19238530090088199542891757945, −5.92452977381891733782531912474, −5.47015142215096301268374242753, −5.23984024455314915212165581436, −4.96069421722156684253306362191, −4.93952840794237129268936575334, −4.76543717699544800769817134698, −4.37976540661138603920334990965, −3.33274884283984404954528243094, −3.23591597422873610430619175012, −2.95137031165480223010839846398, −2.54558009187604436768145381121, −2.38076062677276187021180587857, −1.57290425694890568464325421709, −1.33345990569039309214587107852, −1.24099179305324420132600310347,
1.24099179305324420132600310347, 1.33345990569039309214587107852, 1.57290425694890568464325421709, 2.38076062677276187021180587857, 2.54558009187604436768145381121, 2.95137031165480223010839846398, 3.23591597422873610430619175012, 3.33274884283984404954528243094, 4.37976540661138603920334990965, 4.76543717699544800769817134698, 4.93952840794237129268936575334, 4.96069421722156684253306362191, 5.23984024455314915212165581436, 5.47015142215096301268374242753, 5.92452977381891733782531912474, 6.19238530090088199542891757945, 6.19801910734799788340793746357, 6.32357887775229766665561452861, 6.78319011778848336585267442274, 6.93056979503681191081223231883, 7.26146128510121315295082439118, 7.48309950394572100867730483641, 7.61701511355788899978332756371, 8.303034948489753777165551496494, 8.559946486275999449820491772239
