| L(s) = 1 | + 2-s − 5-s + 7-s − 8-s − 10-s − 11-s + 14-s − 16-s − 22-s + 25-s + 2·29-s + 31-s − 35-s + 40-s + 49-s + 50-s + 2·53-s + 55-s − 56-s + 2·58-s + 2·59-s + 62-s + 64-s − 70-s − 2·73-s − 77-s − 2·79-s + ⋯ |
| L(s) = 1 | + 2-s − 5-s + 7-s − 8-s − 10-s − 11-s + 14-s − 16-s − 22-s + 25-s + 2·29-s + 31-s − 35-s + 40-s + 49-s + 50-s + 2·53-s + 55-s − 56-s + 2·58-s + 2·59-s + 62-s + 64-s − 70-s − 2·73-s − 77-s − 2·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.048890708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048890708\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−11.28897729185349757176938587543, −10.40395227072986269825466495831, −10.29849263335958751713969245095, −9.957655495092429496656075901009, −8.936882728039533519672766205727, −8.764271509874093997665091697378, −8.351435245760952149420637395740, −7.998487181188515708187749811383, −7.49486305848357560215769751303, −6.90776848026876973391307044933, −6.61716070624983645715831749245, −5.69502274832182190560991435890, −5.54109348419150647683992793029, −4.86281071228847127848320295869, −4.51741263336290233158683793598, −4.17084435864189443024042154603, −3.53961642941555128578955342146, −2.68129521586541666852261765784, −2.58308906498773991293265375168, −1.09986560147069669109037785716,
1.09986560147069669109037785716, 2.58308906498773991293265375168, 2.68129521586541666852261765784, 3.53961642941555128578955342146, 4.17084435864189443024042154603, 4.51741263336290233158683793598, 4.86281071228847127848320295869, 5.54109348419150647683992793029, 5.69502274832182190560991435890, 6.61716070624983645715831749245, 6.90776848026876973391307044933, 7.49486305848357560215769751303, 7.998487181188515708187749811383, 8.351435245760952149420637395740, 8.764271509874093997665091697378, 8.936882728039533519672766205727, 9.957655495092429496656075901009, 10.29849263335958751713969245095, 10.40395227072986269825466495831, 11.28897729185349757176938587543
