| L(s) = 1 | − 13-s − 2·19-s + 25-s − 2·37-s − 3·43-s − 49-s + 61-s + 3·67-s − 73-s + 3·79-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 13-s − 2·19-s + 25-s − 2·37-s − 3·43-s − 49-s + 61-s + 3·67-s − 73-s + 3·79-s + 2·97-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9022563762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9022563762\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−8.956347155768019861844335080888, −8.717835917660286676892196492538, −8.652040467403177403114760016448, −7.952547732500360601660244119708, −7.899988869357729301837164397719, −7.22247672072157391668508463573, −6.73317677564875107917777856273, −6.61629568662586173017187274181, −6.42056799225751556685081445375, −5.65695881297572421585454867291, −5.19079921750167853075494526374, −4.81377805780149879784234951135, −4.77565610126035143931323831375, −3.97943034627320081865768966328, −3.52801838285196907328727236870, −3.25102498689477295501650107012, −2.52484782754430229248165302089, −1.94496749112244977968865216055, −1.81980078647027655032016534291, −0.58543185996533308917789117660,
0.58543185996533308917789117660, 1.81980078647027655032016534291, 1.94496749112244977968865216055, 2.52484782754430229248165302089, 3.25102498689477295501650107012, 3.52801838285196907328727236870, 3.97943034627320081865768966328, 4.77565610126035143931323831375, 4.81377805780149879784234951135, 5.19079921750167853075494526374, 5.65695881297572421585454867291, 6.42056799225751556685081445375, 6.61629568662586173017187274181, 6.73317677564875107917777856273, 7.22247672072157391668508463573, 7.899988869357729301837164397719, 7.952547732500360601660244119708, 8.652040467403177403114760016448, 8.717835917660286676892196492538, 8.956347155768019861844335080888
