| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s − 2·9-s + 2·10-s − 8·13-s + 16-s + 6·17-s + 2·18-s − 2·20-s + 3·25-s + 8·26-s + 6·29-s − 32-s − 6·34-s − 2·36-s + 2·37-s + 2·40-s + 6·41-s + 4·45-s − 13·49-s − 3·50-s − 8·52-s + 6·53-s − 6·58-s − 2·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s − 2/3·9-s + 0.632·10-s − 2.21·13-s + 1/4·16-s + 1.45·17-s + 0.471·18-s − 0.447·20-s + 3/5·25-s + 1.56·26-s + 1.11·29-s − 0.176·32-s − 1.02·34-s − 1/3·36-s + 0.328·37-s + 0.316·40-s + 0.937·41-s + 0.596·45-s − 1.85·49-s − 0.424·50-s − 1.10·52-s + 0.824·53-s − 0.787·58-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1095200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
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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
−7.80191348675256968137318821157, −7.47402466508823037518119617944, −7.25190206823653094417112473387, −6.73483784989315938735561487128, −6.05862458218504358555244114858, −5.73299805716885755231121374807, −5.14462937098489056863036116363, −4.60963444019434721197685153570, −4.35398309815315781351355776419, −3.40211614659968515387465276614, −2.99576716331676506161499712038, −2.65209192124627030493117144784, −1.82892205192733241717251603919, −0.856493313433969578668782918754, 0,
0.856493313433969578668782918754, 1.82892205192733241717251603919, 2.65209192124627030493117144784, 2.99576716331676506161499712038, 3.40211614659968515387465276614, 4.35398309815315781351355776419, 4.60963444019434721197685153570, 5.14462937098489056863036116363, 5.73299805716885755231121374807, 6.05862458218504358555244114858, 6.73483784989315938735561487128, 7.25190206823653094417112473387, 7.47402466508823037518119617944, 7.80191348675256968137318821157
