| L(s) = 1 | + 2·2-s + 2·4-s + 2·5-s − 3·9-s + 4·10-s − 4·16-s − 6·18-s − 2·19-s + 4·20-s + 7·23-s − 6·25-s − 4·29-s − 8·32-s − 6·36-s − 4·38-s + 10·43-s − 6·45-s + 14·46-s + 8·47-s − 10·49-s − 12·50-s + 6·53-s − 8·58-s − 8·64-s − 14·67-s + 16·71-s − 4·73-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.894·5-s − 9-s + 1.26·10-s − 16-s − 1.41·18-s − 0.458·19-s + 0.894·20-s + 1.45·23-s − 6/5·25-s − 0.742·29-s − 1.41·32-s − 36-s − 0.648·38-s + 1.52·43-s − 0.894·45-s + 2.06·46-s + 1.16·47-s − 1.42·49-s − 1.69·50-s + 0.824·53-s − 1.05·58-s − 64-s − 1.71·67-s + 1.89·71-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13248 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.119988428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.119988428\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 8 T + p T^{2} ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−11.26173646234662020739266148196, −11.00415419728308459505175138458, −10.31669178314621384742420826035, −9.411283824513505183274863624268, −9.219939562777853710111455204207, −8.529001155147874615033175513395, −7.71716791211264924599187845314, −6.97932297081441080255900950715, −6.26979014646319204285171040318, −5.74095421692274996052286327147, −5.41437444373033840627514220133, −4.58068140149052752719358631104, −3.77967386403094492256097723236, −2.91848441729996005192586996444, −2.14839899127527159079475191622,
2.14839899127527159079475191622, 2.91848441729996005192586996444, 3.77967386403094492256097723236, 4.58068140149052752719358631104, 5.41437444373033840627514220133, 5.74095421692274996052286327147, 6.26979014646319204285171040318, 6.97932297081441080255900950715, 7.71716791211264924599187845314, 8.529001155147874615033175513395, 9.219939562777853710111455204207, 9.411283824513505183274863624268, 10.31669178314621384742420826035, 11.00415419728308459505175138458, 11.26173646234662020739266148196
