| L(s) = 1 | − 4-s + 5-s − 7-s − 9-s + 11-s − 2·17-s − 19-s − 20-s + 23-s + 25-s + 28-s − 35-s + 36-s − 2·43-s − 44-s − 45-s + 47-s + 55-s + 61-s + 63-s + 64-s + 2·68-s + 73-s + 76-s − 77-s − 2·83-s − 2·85-s + ⋯ |
| L(s) = 1 | − 4-s + 5-s − 7-s − 9-s + 11-s − 2·17-s − 19-s − 20-s + 23-s + 25-s + 28-s − 35-s + 36-s − 2·43-s − 44-s − 45-s + 47-s + 55-s + 61-s + 63-s + 64-s + 2·68-s + 73-s + 76-s − 77-s − 2·83-s − 2·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17689 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3307553836\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3307553836\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + T + T^{2} \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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
−13.51536939115785482243110465220, −13.42880854240873474245589123863, −12.82004215862498781437345667003, −12.61363320452011912066508697198, −11.46539040593524054685473501032, −11.42606310251343268145856726421, −10.59528197589011550738162946023, −10.10117837795661673178713172310, −9.311234802124883832962565317195, −9.239794672657171368592271462543, −8.586299636721084614348707081449, −8.448405537237828755980821676130, −6.94213003987707110179046690510, −6.69334998687290640538404127101, −6.18638201546974727841309198593, −5.42331299502316119487515539401, −4.70809796419164559349036462342, −4.06434645796077638680479019311, −3.09997349348192763214703328180, −2.16502060743839806249047133845,
2.16502060743839806249047133845, 3.09997349348192763214703328180, 4.06434645796077638680479019311, 4.70809796419164559349036462342, 5.42331299502316119487515539401, 6.18638201546974727841309198593, 6.69334998687290640538404127101, 6.94213003987707110179046690510, 8.448405537237828755980821676130, 8.586299636721084614348707081449, 9.239794672657171368592271462543, 9.311234802124883832962565317195, 10.10117837795661673178713172310, 10.59528197589011550738162946023, 11.42606310251343268145856726421, 11.46539040593524054685473501032, 12.61363320452011912066508697198, 12.82004215862498781437345667003, 13.42880854240873474245589123863, 13.51536939115785482243110465220
