| L(s) = 1 | − 4·4-s + 3·5-s − 2·7-s + 2·11-s − 3·13-s + 12·16-s − 9·17-s + 8·19-s − 12·20-s − 3·23-s + 25-s + 8·28-s − 3·31-s − 6·35-s − 12·37-s + 41-s + 8·43-s − 8·44-s + 18·47-s + 3·49-s + 12·52-s + 22·53-s + 6·55-s + 13·59-s − 15·61-s − 32·64-s − 9·65-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.34·5-s − 0.755·7-s + 0.603·11-s − 0.832·13-s + 3·16-s − 2.18·17-s + 1.83·19-s − 2.68·20-s − 0.625·23-s + 1/5·25-s + 1.51·28-s − 0.538·31-s − 1.01·35-s − 1.97·37-s + 0.156·41-s + 1.21·43-s − 1.20·44-s + 2.62·47-s + 3/7·49-s + 1.66·52-s + 3.02·53-s + 0.809·55-s + 1.69·59-s − 1.92·61-s − 4·64-s − 1.11·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64016001 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64016001 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 127 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 24 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T + 26 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 93 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 158 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 13 T + 156 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 15 T + 174 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 206 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 13 T + 184 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 13 T + 162 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 15 T + 116 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T + 78 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 90 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−7.44136674505953262089157562709, −7.38429401110820330868565780220, −6.89606601032260173011221489390, −6.82313560103423260722374516800, −5.92941709857699826838269339897, −5.87276776489624215003881744529, −5.53460947939178620504082036143, −5.38780045626242777932726458878, −4.85778870256030746362785242912, −4.44652914758158435316360051376, −4.06561941990325323286361794382, −3.86119767126853105853179703567, −3.51393760001950688532530569394, −2.78875097811252789394803703666, −2.44418355168446324462678173059, −2.13895672732756781957870508733, −1.22037487584328205368848134808, −1.11554157380032237055958190698, 0, 0,
1.11554157380032237055958190698, 1.22037487584328205368848134808, 2.13895672732756781957870508733, 2.44418355168446324462678173059, 2.78875097811252789394803703666, 3.51393760001950688532530569394, 3.86119767126853105853179703567, 4.06561941990325323286361794382, 4.44652914758158435316360051376, 4.85778870256030746362785242912, 5.38780045626242777932726458878, 5.53460947939178620504082036143, 5.87276776489624215003881744529, 5.92941709857699826838269339897, 6.82313560103423260722374516800, 6.89606601032260173011221489390, 7.38429401110820330868565780220, 7.44136674505953262089157562709
