L(s) = 1 | − 4·2-s + 4·3-s + 10·4-s − 2·5-s − 16·6-s − 20·8-s + 10·9-s + 8·10-s − 4·11-s + 40·12-s − 8·15-s + 35·16-s − 2·17-s − 40·18-s − 2·19-s − 20·20-s + 16·22-s − 80·24-s + 25-s + 32·27-s − 8·29-s + 32·30-s + 12·31-s − 56·32-s − 16·33-s + 8·34-s + 100·36-s + ⋯ |
L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s − 0.894·5-s − 6.53·6-s − 7.07·8-s + 10/3·9-s + 2.52·10-s − 1.20·11-s + 11.5·12-s − 2.06·15-s + 35/4·16-s − 0.485·17-s − 9.42·18-s − 0.458·19-s − 4.47·20-s + 3.41·22-s − 16.3·24-s + 1/5·25-s + 6.15·27-s − 1.48·29-s + 5.84·30-s + 2.15·31-s − 9.89·32-s − 2.78·33-s + 1.37·34-s + 50/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 43^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8934924097\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8934924097\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 6 T + 55 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
good | 3 | $C_2^2$ | \( ( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 4 T^{2} - 33 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 16 T^{2} + 87 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 2 T - 21 T^{2} - 18 T^{3} + 268 T^{4} - 18 p T^{5} - 21 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 25 T^{2} - 18 T^{3} + 404 T^{4} - 18 p T^{5} - 25 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 8 T + 30 T^{2} - 192 T^{3} - 1541 T^{4} - 192 p T^{5} + 30 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 56 T^{2} - 312 T^{3} + 2319 T^{4} - 312 p T^{5} + 56 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 10 T + 67 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T - 18 T^{2} + 192 T^{3} + 523 T^{4} + 192 p T^{5} - 18 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 14 T + 157 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 4 T - 100 T^{2} + 24 T^{3} + 8759 T^{4} + 24 p T^{5} - 100 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T + 29 T^{2} - 318 T^{3} - 4132 T^{4} - 318 p T^{5} + 29 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 16 T + 60 T^{2} + 864 T^{3} + 15199 T^{4} + 864 p T^{5} + 60 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 6 T - 109 T^{2} - 6 T^{3} + 13068 T^{4} - 6 p T^{5} - 109 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^3$ | \( 1 - 148 T^{2} + 15663 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 14 T + 233 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.235889839083139328702689487469, −7.81695821788501219381455712635, −7.81582190248355035774060727475, −7.66815242656322134461996314839, −7.29364182955520077699804451527, −7.05148519318817135856948917332, −6.78726608604236551978671333907, −6.65885077754414738031865719978, −6.63491692241535726600897667554, −5.90080480462899670365842096180, −5.81478389051007572530365409793, −5.15623967441424998715259753486, −5.05964591032835850956600539635, −4.69091655557659445931945952318, −4.14387396974331990472656400848, −4.06273992933967226429663978719, −3.40227139491276951917314056207, −3.25925852659407306975816869845, −3.17973545969477345084686985716, −2.45802838601061768796467830684, −2.35669848090057977057481462836, −2.27076426258596303575280060471, −1.62719435781309491401221041254, −1.15211250984936923393592794257, −0.50871132316289690088736390091,
0.50871132316289690088736390091, 1.15211250984936923393592794257, 1.62719435781309491401221041254, 2.27076426258596303575280060471, 2.35669848090057977057481462836, 2.45802838601061768796467830684, 3.17973545969477345084686985716, 3.25925852659407306975816869845, 3.40227139491276951917314056207, 4.06273992933967226429663978719, 4.14387396974331990472656400848, 4.69091655557659445931945952318, 5.05964591032835850956600539635, 5.15623967441424998715259753486, 5.81478389051007572530365409793, 5.90080480462899670365842096180, 6.63491692241535726600897667554, 6.65885077754414738031865719978, 6.78726608604236551978671333907, 7.05148519318817135856948917332, 7.29364182955520077699804451527, 7.66815242656322134461996314839, 7.81582190248355035774060727475, 7.81695821788501219381455712635, 8.235889839083139328702689487469