| L(s) = 1 | − 2·5-s + 9-s + 2·17-s + 25-s + 2·37-s − 2·45-s + 2·53-s + 2·61-s − 2·73-s − 4·85-s − 2·89-s − 2·101-s − 2·109-s + 121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
| L(s) = 1 | − 2·5-s + 9-s + 2·17-s + 25-s + 2·37-s − 2·45-s + 2·53-s + 2·61-s − 2·73-s − 4·85-s − 2·89-s − 2·101-s − 2·109-s + 121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9834496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.048253563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.048253563\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.030759261522037662342844967463, −8.357460455372478187408822827656, −8.316444711296946184129156834749, −7.80540314590740505037219241358, −7.70226811793866219209590230907, −7.27223805000644486927166408140, −6.93944843643992928231718063747, −6.68783916127040591966224004624, −5.78784786065924833322688979849, −5.73619881553082902125118001135, −5.27966987231398499456077575938, −4.62750789169649019052847907625, −4.11481905407659555370901207468, −4.07432788202454569792710585547, −3.76910602147519921733374034967, −2.96498103787881621447586540196, −2.90712141387924920560531459171, −1.98108995431133057429718750357, −1.25516289156321947270923558117, −0.72629814496278636996476048835,
0.72629814496278636996476048835, 1.25516289156321947270923558117, 1.98108995431133057429718750357, 2.90712141387924920560531459171, 2.96498103787881621447586540196, 3.76910602147519921733374034967, 4.07432788202454569792710585547, 4.11481905407659555370901207468, 4.62750789169649019052847907625, 5.27966987231398499456077575938, 5.73619881553082902125118001135, 5.78784786065924833322688979849, 6.68783916127040591966224004624, 6.93944843643992928231718063747, 7.27223805000644486927166408140, 7.70226811793866219209590230907, 7.80540314590740505037219241358, 8.316444711296946184129156834749, 8.357460455372478187408822827656, 9.030759261522037662342844967463
