| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 5·9-s − 2·10-s + 4·11-s + 13-s − 16-s − 2·17-s + 5·18-s − 8·19-s − 2·20-s − 4·22-s + 4·23-s − 3·25-s − 26-s − 5·32-s + 2·34-s + 5·36-s + 2·37-s + 8·38-s + 6·40-s − 4·44-s − 10·45-s − 4·46-s − 9·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 5/3·9-s − 0.632·10-s + 1.20·11-s + 0.277·13-s − 1/4·16-s − 0.485·17-s + 1.17·18-s − 1.83·19-s − 0.447·20-s − 0.852·22-s + 0.834·23-s − 3/5·25-s − 0.196·26-s − 0.883·32-s + 0.342·34-s + 5/6·36-s + 0.328·37-s + 1.29·38-s + 0.948·40-s − 0.603·44-s − 1.49·45-s − 0.589·46-s − 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_1$ | \( 1 - T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.001040525976181479884005598799, −8.880763265166357867059686877296, −8.260593762335526283446182342044, −7.915079971859779575507889375384, −7.12304749717617036421257115984, −6.50950585285465347204220183121, −6.06012504381392955850336440028, −5.80237067447431834670967262969, −4.93155463160558538357086907666, −4.47504998423005343412551843659, −3.80292887542373980459274005614, −3.02427326417789224299678820422, −2.17462965289367376859494732042, −1.45036696602345253303979075739, 0,
1.45036696602345253303979075739, 2.17462965289367376859494732042, 3.02427326417789224299678820422, 3.80292887542373980459274005614, 4.47504998423005343412551843659, 4.93155463160558538357086907666, 5.80237067447431834670967262969, 6.06012504381392955850336440028, 6.50950585285465347204220183121, 7.12304749717617036421257115984, 7.915079971859779575507889375384, 8.260593762335526283446182342044, 8.880763265166357867059686877296, 9.001040525976181479884005598799
