| L(s) = 1 | + 7-s + 4·13-s − 19-s − 25-s − 31-s − 2·37-s + 2·43-s + 61-s + 2·67-s + 73-s + 2·79-s + 4·91-s − 2·97-s + 2·103-s + 109-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
| L(s) = 1 | + 7-s + 4·13-s − 19-s − 25-s − 31-s − 2·37-s + 2·43-s + 61-s + 2·67-s + 73-s + 2·79-s + 4·91-s − 2·97-s + 2·103-s + 109-s − 121-s + 127-s + 131-s − 133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.970726242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.970726242\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−8.815271575623316302579779526192, −8.636490401739131474921612612021, −8.470877040860213199680862253172, −8.128001158619050544783132378622, −7.67341998901374022765975351397, −7.30878386537984907077780515697, −6.65823201422331666216501939357, −6.43994571219116060256033335697, −6.03357412953258267668983010155, −5.78913769048769684396343853399, −5.25370995451373661846185443757, −4.98411873408828019884347305389, −4.19258181092855002355422033845, −3.92651313435897278522217883641, −3.52997627047824460349164794222, −3.49107410634711396724391598675, −2.30514510687235254119250570635, −2.05408685603753019419363762605, −1.37554072832928014524620016654, −1.02668165045805675640069673409,
1.02668165045805675640069673409, 1.37554072832928014524620016654, 2.05408685603753019419363762605, 2.30514510687235254119250570635, 3.49107410634711396724391598675, 3.52997627047824460349164794222, 3.92651313435897278522217883641, 4.19258181092855002355422033845, 4.98411873408828019884347305389, 5.25370995451373661846185443757, 5.78913769048769684396343853399, 6.03357412953258267668983010155, 6.43994571219116060256033335697, 6.65823201422331666216501939357, 7.30878386537984907077780515697, 7.67341998901374022765975351397, 8.128001158619050544783132378622, 8.470877040860213199680862253172, 8.636490401739131474921612612021, 8.815271575623316302579779526192
