| L(s) = 1 | − 2-s + 4·4-s + 12·5-s − 7·7-s − 11·8-s − 3·9-s − 12·10-s + 32·11-s + 13·13-s + 7·14-s + 11·16-s − 18·17-s + 3·18-s + 48·20-s − 32·22-s − 4·23-s + 71·25-s − 13·26-s − 28·28-s + 38·29-s − 36·31-s − 44·32-s + 18·34-s − 84·35-s − 12·36-s − 17·37-s − 132·40-s + ⋯ |
| L(s) = 1 | − 1/2·2-s + 4-s + 12/5·5-s − 7-s − 1.37·8-s − 1/3·9-s − 6/5·10-s + 2.90·11-s + 13-s + 1/2·14-s + 0.687·16-s − 1.05·17-s + 1/6·18-s + 12/5·20-s − 1.45·22-s − 0.173·23-s + 2.83·25-s − 1/2·26-s − 28-s + 1.31·29-s − 1.16·31-s − 1.37·32-s + 9/17·34-s − 2.39·35-s − 1/3·36-s − 0.459·37-s − 3.29·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.138401277\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.138401277\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - 3 T^{2} + p^{2} T^{3} + p^{4} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 12 T + 73 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 18 T + 397 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 215 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 513 T^{2} + 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 38 T + 603 T^{2} - 38 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 36 T + 1393 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 17 T - 1080 T^{2} + 17 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 72 T + 3409 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 29 T - 1008 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 54 T + 3181 T^{2} - 54 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 40 T - 1209 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 114 T + 7813 T^{2} - 114 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )( 1 + 47 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 38 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 122 T + 9843 T^{2} - 122 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 93 T + 8212 T^{2} - 93 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 110 T + 5859 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 13586 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 252 T + 29089 T^{2} - 252 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 153 T + 17212 T^{2} - 153 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−11.95902382179321905004390405618, −11.22488167505585336184062649579, −11.14619877834328911531036653100, −10.28573416593000355054848830573, −9.867024544273015349200434746461, −9.572923666257070075663191668414, −9.076977321844362311283762526447, −8.731960276423561460516502352525, −8.586511599161917377986228698299, −7.09557034294648326660543146470, −6.67384033948967563585832636561, −6.39488867995826833621768725262, −6.25334285950647478653503954270, −5.74129387568169768265638655520, −4.92894872059187100032111799361, −3.60495722267018392142778081937, −3.47258895153008436746743151420, −2.21552553649496405938861855446, −1.96543140585216920815810926277, −1.02784823714545493163810671067,
1.02784823714545493163810671067, 1.96543140585216920815810926277, 2.21552553649496405938861855446, 3.47258895153008436746743151420, 3.60495722267018392142778081937, 4.92894872059187100032111799361, 5.74129387568169768265638655520, 6.25334285950647478653503954270, 6.39488867995826833621768725262, 6.67384033948967563585832636561, 7.09557034294648326660543146470, 8.586511599161917377986228698299, 8.731960276423561460516502352525, 9.076977321844362311283762526447, 9.572923666257070075663191668414, 9.867024544273015349200434746461, 10.28573416593000355054848830573, 11.14619877834328911531036653100, 11.22488167505585336184062649579, 11.95902382179321905004390405618
