L(s) = 1 | + 2·2-s + 3·4-s + 5-s + 4·8-s + 2·10-s + 5·16-s + 2·19-s + 3·20-s − 23-s − 3·31-s + 6·32-s + 4·38-s + 4·40-s − 2·46-s − 2·47-s + 49-s − 2·53-s + 3·61-s − 6·62-s + 7·64-s + 6·76-s − 3·79-s + 5·80-s − 3·83-s − 3·92-s − 4·94-s + 2·95-s + ⋯ |
L(s) = 1 | + 2·2-s + 3·4-s + 5-s + 4·8-s + 2·10-s + 5·16-s + 2·19-s + 3·20-s − 23-s − 3·31-s + 6·32-s + 4·38-s + 4·40-s − 2·46-s − 2·47-s + 49-s − 2·53-s + 3·61-s − 6·62-s + 7·64-s + 6·76-s − 3·79-s + 5·80-s − 3·83-s − 3·92-s − 4·94-s + 2·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(7.279079742\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.279079742\) |
\(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−8.985415510924040841652380932550, −8.644431530835748219528419054636, −8.013046953223404996845233618761, −7.71548024687354504338870134728, −7.22920137302163859514935722425, −7.19342726863504151283290926730, −6.58598200328578019443890668890, −6.27324337157929223761397343478, −5.75645420659034615353133038412, −5.57426773100597854767857035521, −5.15994669509305035617800269702, −5.13865282203846834763251167716, −4.25294712941792186571947289940, −4.02295480227100309543703328782, −3.55573528832098950261198487490, −3.12388743892840018489691919905, −2.74207081252684412975609324357, −2.15725181972387401659466000115, −1.56808751671950202533415454776, −1.46879081297093408809818655709,
1.46879081297093408809818655709, 1.56808751671950202533415454776, 2.15725181972387401659466000115, 2.74207081252684412975609324357, 3.12388743892840018489691919905, 3.55573528832098950261198487490, 4.02295480227100309543703328782, 4.25294712941792186571947289940, 5.13865282203846834763251167716, 5.15994669509305035617800269702, 5.57426773100597854767857035521, 5.75645420659034615353133038412, 6.27324337157929223761397343478, 6.58598200328578019443890668890, 7.19342726863504151283290926730, 7.22920137302163859514935722425, 7.71548024687354504338870134728, 8.013046953223404996845233618761, 8.644431530835748219528419054636, 8.985415510924040841652380932550