| L(s) = 1 | − 6·9-s − 6·13-s − 4·17-s − 6·25-s + 20·29-s + 14·49-s + 28·53-s − 20·61-s + 27·81-s − 4·101-s + 28·113-s + 36·117-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s + 23·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·9-s − 1.66·13-s − 0.970·17-s − 6/5·25-s + 3.71·29-s + 2·49-s + 3.84·53-s − 2.56·61-s + 3·81-s − 0.398·101-s + 2.63·113-s + 3.32·117-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9534167305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9534167305\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−11.55787075739832438918349982840, −10.94623992581870519931806295802, −10.57524265039742480878316034732, −9.980054978256597878816658108354, −9.851700634749097054144291123639, −8.835093893370694411023005394471, −8.775739352324658909156762102108, −8.473765674485950046188673942610, −7.70294448963979100495083754388, −7.33458006870500264489307943622, −6.73761233265253247376963409325, −6.14192674121894524829240884087, −5.80091516039162686028993999232, −5.14439042862379674767591967660, −4.67785442856069665614237807109, −4.12631740983543554866834252820, −3.18000753413123720366118804601, −2.44773161809234098578861073100, −2.42921245897540854870854579688, −0.60630896463668980869045042343,
0.60630896463668980869045042343, 2.42921245897540854870854579688, 2.44773161809234098578861073100, 3.18000753413123720366118804601, 4.12631740983543554866834252820, 4.67785442856069665614237807109, 5.14439042862379674767591967660, 5.80091516039162686028993999232, 6.14192674121894524829240884087, 6.73761233265253247376963409325, 7.33458006870500264489307943622, 7.70294448963979100495083754388, 8.473765674485950046188673942610, 8.775739352324658909156762102108, 8.835093893370694411023005394471, 9.851700634749097054144291123639, 9.980054978256597878816658108354, 10.57524265039742480878316034732, 10.94623992581870519931806295802, 11.55787075739832438918349982840
