| L(s) = 1 | − 2-s − 3-s − 4·5-s + 6-s − 5·7-s + 8-s + 4·10-s + 5·11-s + 4·13-s + 5·14-s + 4·15-s − 16-s − 3·17-s − 6·19-s + 5·21-s − 5·22-s − 23-s − 24-s + 5·25-s − 4·26-s + 27-s + 2·29-s − 4·30-s − 10·31-s − 5·33-s + 3·34-s + 20·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1.78·5-s + 0.408·6-s − 1.88·7-s + 0.353·8-s + 1.26·10-s + 1.50·11-s + 1.10·13-s + 1.33·14-s + 1.03·15-s − 1/4·16-s − 0.727·17-s − 1.37·19-s + 1.09·21-s − 1.06·22-s − 0.208·23-s − 0.204·24-s + 25-s − 0.784·26-s + 0.192·27-s + 0.371·29-s − 0.730·30-s − 1.79·31-s − 0.870·33-s + 0.514·34-s + 3.38·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 23 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.932093256444200154042601618704, −9.157975200493361443806801915097, −8.869773360087666613178326351112, −8.756234144530702976368701290316, −8.366175093264991259531667379029, −7.61257972657227227748642201909, −7.28261785997490355654126292705, −6.82807790916402239702775605943, −6.53651493955121722141094445593, −5.99114551668135510457520251319, −5.85625664898240530427688871595, −4.77158522227531970868062298972, −4.27672242853765579951553819433, −3.93150132516452500613502136697, −3.52982060627537198567585272887, −3.17405408528451616005490216768, −2.08784141558348800551252284105, −1.18982027257948313960052765455, 0, 0,
1.18982027257948313960052765455, 2.08784141558348800551252284105, 3.17405408528451616005490216768, 3.52982060627537198567585272887, 3.93150132516452500613502136697, 4.27672242853765579951553819433, 4.77158522227531970868062298972, 5.85625664898240530427688871595, 5.99114551668135510457520251319, 6.53651493955121722141094445593, 6.82807790916402239702775605943, 7.28261785997490355654126292705, 7.61257972657227227748642201909, 8.366175093264991259531667379029, 8.756234144530702976368701290316, 8.869773360087666613178326351112, 9.157975200493361443806801915097, 9.932093256444200154042601618704
