| L(s) = 1 | − 3-s + 2·4-s + 7-s + 2·11-s − 2·12-s − 6·13-s − 2·17-s − 3·19-s − 21-s − 23-s + 5·25-s + 27-s + 2·28-s + 12·29-s + 3·31-s − 2·33-s + 3·37-s + 6·39-s + 6·43-s + 4·44-s + 10·47-s − 6·49-s + 2·51-s − 12·52-s + 4·53-s + 3·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s + 0.377·7-s + 0.603·11-s − 0.577·12-s − 1.66·13-s − 0.485·17-s − 0.688·19-s − 0.218·21-s − 0.208·23-s + 25-s + 0.192·27-s + 0.377·28-s + 2.22·29-s + 0.538·31-s − 0.348·33-s + 0.493·37-s + 0.960·39-s + 0.914·43-s + 0.603·44-s + 1.45·47-s − 6/7·49-s + 0.280·51-s − 1.66·52-s + 0.549·53-s + 0.397·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 233289 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.719144322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.719144322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| 23 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 3 T - 10 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + T - 66 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−11.47336983773104904235557683552, −10.68483642553712540580669288659, −10.42045810761469081055717817051, −10.16436477849446367153882541146, −9.256790894451603555770324442040, −9.198462570764373662748236291977, −8.389462246331985922386263755143, −8.068739417884246567022959148542, −7.42692095864344879048971388049, −6.86600108735784035500470346867, −6.74025870014379059487372318660, −6.23477990751254073746139050507, −5.63648180212877538251114042523, −4.99980390802012691916442982875, −4.47962722379806541495878870072, −4.20401699536831176714123535759, −2.94760024981985077568430727468, −2.59920772360186432263395704160, −1.96328165780901198634610387737, −0.830448254714121241522396090724,
0.830448254714121241522396090724, 1.96328165780901198634610387737, 2.59920772360186432263395704160, 2.94760024981985077568430727468, 4.20401699536831176714123535759, 4.47962722379806541495878870072, 4.99980390802012691916442982875, 5.63648180212877538251114042523, 6.23477990751254073746139050507, 6.74025870014379059487372318660, 6.86600108735784035500470346867, 7.42692095864344879048971388049, 8.068739417884246567022959148542, 8.389462246331985922386263755143, 9.198462570764373662748236291977, 9.256790894451603555770324442040, 10.16436477849446367153882541146, 10.42045810761469081055717817051, 10.68483642553712540580669288659, 11.47336983773104904235557683552
