| L(s) = 1 | + 2·5-s + 9-s + 3·25-s − 2·29-s − 2·41-s + 2·45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s + 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 2·5-s + 9-s + 3·25-s − 2·29-s − 2·41-s + 2·45-s − 2·61-s + 2·89-s + 2·101-s + 2·109-s + 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.483664859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.483664859\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.956108210562206209990784044381, −8.601377427770113582300161973016, −8.128725964837760640838111438724, −7.59514839199470243006805898537, −7.33118290207869700134588603143, −6.96165348348636394809829758425, −6.57040191620155999160882826952, −6.26634060818155616380010169127, −5.72001267061123211093105438507, −5.70779353000963306177534561303, −5.07946398376705954394716485155, −4.75322985178008914011936985210, −4.46792955475968031660661767907, −3.79898656175543985972988257280, −3.18649694421326464983934446220, −3.12591970725538270566900339192, −2.14305379732536993395158047610, −1.91424914110462113324793165761, −1.68155899929199723058289274410, −0.915010044279353065802582012378,
0.915010044279353065802582012378, 1.68155899929199723058289274410, 1.91424914110462113324793165761, 2.14305379732536993395158047610, 3.12591970725538270566900339192, 3.18649694421326464983934446220, 3.79898656175543985972988257280, 4.46792955475968031660661767907, 4.75322985178008914011936985210, 5.07946398376705954394716485155, 5.70779353000963306177534561303, 5.72001267061123211093105438507, 6.26634060818155616380010169127, 6.57040191620155999160882826952, 6.96165348348636394809829758425, 7.33118290207869700134588603143, 7.59514839199470243006805898537, 8.128725964837760640838111438724, 8.601377427770113582300161973016, 8.956108210562206209990784044381
