| L(s) = 1 | − 3-s + 2·5-s − 2·7-s − 11-s + 7·13-s − 2·15-s − 17-s + 4·19-s + 2·21-s − 2·23-s − 7·25-s + 27-s − 3·29-s + 12·31-s + 33-s − 4·35-s + 7·37-s − 7·39-s + 11·41-s + 4·43-s + 24·47-s + 7·49-s + 51-s + 2·53-s − 2·55-s − 4·57-s + 14·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s − 0.301·11-s + 1.94·13-s − 0.516·15-s − 0.242·17-s + 0.917·19-s + 0.436·21-s − 0.417·23-s − 7/5·25-s + 0.192·27-s − 0.557·29-s + 2.15·31-s + 0.174·33-s − 0.676·35-s + 1.15·37-s − 1.12·39-s + 1.71·41-s + 0.609·43-s + 3.50·47-s + 49-s + 0.140·51-s + 0.274·53-s − 0.269·55-s − 0.529·57-s + 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2944656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.653478694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.653478694\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 11 T + 80 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 10 T + 29 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 6 T - 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.455058750275134078115133466743, −9.287150987170865557225447602838, −8.821056946756529719788004111802, −8.400928367338268580629682729904, −7.78607526108738667186961741841, −7.72559188958323324644187687498, −7.00717465938425997965815148928, −6.61040334304226897250484592180, −6.09429470697490922905200353412, −5.95542784415323005695035991402, −5.54273866654926758356171137474, −5.44625117296806816937470479959, −4.31865984154631860570547162902, −4.19431796067402910915339185650, −3.74583758553481223299268070821, −3.07013670008561635873209680419, −2.40688206579192990137759412535, −2.20334272310921005442994013193, −0.993453502620271445256633462697, −0.845016722722214414069878673251,
0.845016722722214414069878673251, 0.993453502620271445256633462697, 2.20334272310921005442994013193, 2.40688206579192990137759412535, 3.07013670008561635873209680419, 3.74583758553481223299268070821, 4.19431796067402910915339185650, 4.31865984154631860570547162902, 5.44625117296806816937470479959, 5.54273866654926758356171137474, 5.95542784415323005695035991402, 6.09429470697490922905200353412, 6.61040334304226897250484592180, 7.00717465938425997965815148928, 7.72559188958323324644187687498, 7.78607526108738667186961741841, 8.400928367338268580629682729904, 8.821056946756529719788004111802, 9.287150987170865557225447602838, 9.455058750275134078115133466743
