| L(s) = 1 | + 2-s − 4-s − 2·5-s − 3·8-s + 9-s − 2·10-s − 8·13-s − 16-s + 4·17-s + 18-s + 2·20-s + 3·25-s − 8·26-s + 8·29-s + 5·32-s + 4·34-s − 36-s + 6·40-s − 2·45-s − 10·49-s + 3·50-s + 8·52-s − 4·53-s + 8·58-s + 4·61-s + 7·64-s + 16·65-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 2.21·13-s − 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.447·20-s + 3/5·25-s − 1.56·26-s + 1.48·29-s + 0.883·32-s + 0.685·34-s − 1/6·36-s + 0.948·40-s − 0.298·45-s − 1.42·49-s + 0.424·50-s + 1.10·52-s − 0.549·53-s + 1.05·58-s + 0.512·61-s + 7/8·64-s + 1.98·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
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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
−7.69451426376713427108997802036, −7.28034875915901512150776673175, −7.07341113368588561296966396700, −6.27483396251328001307537385761, −6.11724584120824685520629468061, −5.15467699824668629469430799137, −5.07872552787716309543367621064, −4.69308900565895276645745276376, −4.19187417186236300755390321977, −3.72176395363745906322596037322, −2.93342543120652714539614603200, −2.93269055521134332794719136670, −1.97779909070487265858566219387, −0.895818257201027131012503394679, 0,
0.895818257201027131012503394679, 1.97779909070487265858566219387, 2.93269055521134332794719136670, 2.93342543120652714539614603200, 3.72176395363745906322596037322, 4.19187417186236300755390321977, 4.69308900565895276645745276376, 5.07872552787716309543367621064, 5.15467699824668629469430799137, 6.11724584120824685520629468061, 6.27483396251328001307537385761, 7.07341113368588561296966396700, 7.28034875915901512150776673175, 7.69451426376713427108997802036
