| L(s) = 1 | − 2·3-s − 4-s − 2·5-s − 4·7-s + 3·9-s − 2·11-s + 2·12-s − 2·13-s + 4·15-s − 3·16-s + 10·17-s − 8·19-s + 2·20-s + 8·21-s − 4·23-s − 4·25-s − 4·27-s + 4·28-s + 2·29-s − 6·31-s + 4·33-s + 8·35-s − 3·36-s − 4·37-s + 4·39-s − 4·41-s − 2·43-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 0.894·5-s − 1.51·7-s + 9-s − 0.603·11-s + 0.577·12-s − 0.554·13-s + 1.03·15-s − 3/4·16-s + 2.42·17-s − 1.83·19-s + 0.447·20-s + 1.74·21-s − 0.834·23-s − 4/5·25-s − 0.769·27-s + 0.755·28-s + 0.371·29-s − 1.07·31-s + 0.696·33-s + 1.35·35-s − 1/2·36-s − 0.657·37-s + 0.640·39-s − 0.624·41-s − 0.304·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 10 T + 56 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 32 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 44 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 86 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 20 T + 206 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 156 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 108 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 14 T + 152 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.98233132089374916908110440536, −10.31850574761818891989063494270, −10.09122356878340641242633577971, −9.887918448489030988404690435313, −9.140383791686041291828294967045, −8.810786877525986769989149208569, −7.943656972539248090553778634377, −7.73888656519421108670134539450, −7.24340841132030452266532747571, −6.62780667496321996725889291267, −6.08334615241733373529407954872, −5.85709950063880528463459733171, −5.11423171313381413795979310471, −4.64213184103502313720127885293, −3.97710819556428815156256206048, −3.55474922112396448198929899788, −2.87032313039725980496591990680, −1.73227011454809870487556010975, 0, 0,
1.73227011454809870487556010975, 2.87032313039725980496591990680, 3.55474922112396448198929899788, 3.97710819556428815156256206048, 4.64213184103502313720127885293, 5.11423171313381413795979310471, 5.85709950063880528463459733171, 6.08334615241733373529407954872, 6.62780667496321996725889291267, 7.24340841132030452266532747571, 7.73888656519421108670134539450, 7.943656972539248090553778634377, 8.810786877525986769989149208569, 9.140383791686041291828294967045, 9.887918448489030988404690435313, 10.09122356878340641242633577971, 10.31850574761818891989063494270, 10.98233132089374916908110440536
