| L(s) = 1 | − 3·2-s + 2·3-s + 4·4-s − 5·5-s − 6·6-s + 2·7-s − 3·8-s + 3·9-s + 15·10-s − 2·11-s + 8·12-s − 7·13-s − 6·14-s − 10·15-s + 3·16-s − 2·17-s − 9·18-s − 6·19-s − 20·20-s + 4·21-s + 6·22-s − 2·23-s − 6·24-s + 10·25-s + 21·26-s + 4·27-s + 8·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 1.15·3-s + 2·4-s − 2.23·5-s − 2.44·6-s + 0.755·7-s − 1.06·8-s + 9-s + 4.74·10-s − 0.603·11-s + 2.30·12-s − 1.94·13-s − 1.60·14-s − 2.58·15-s + 3/4·16-s − 0.485·17-s − 2.12·18-s − 1.37·19-s − 4.47·20-s + 0.872·21-s + 1.27·22-s − 0.417·23-s − 1.22·24-s + 2·25-s + 4.11·26-s + 0.769·27-s + 1.51·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 3 p T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + p T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T + 61 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 74 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 125 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 19 T + 197 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T - 27 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 15 T + 179 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 19 T + 201 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 163 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 13 T + 209 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−10.43343526796892694157864761346, −10.39823081149817439046632654717, −9.578180479679418414146930744942, −9.327444753066146962134264057681, −8.777510517566518098706293840748, −8.639076508177464065000002641327, −7.87526213274663625327873522313, −7.86935605681606903657724655349, −7.42769423515147453862951066724, −7.40093200900812343193993508106, −6.56845343610710827421950839293, −5.57563154157219914651930065485, −4.66043629966342784974621871937, −4.50426503132686146127872626357, −3.69656774805218198534431793330, −3.24229584203906335485827551561, −2.23088730330174361858629621432, −1.78219363577744059942077016856, 0, 0,
1.78219363577744059942077016856, 2.23088730330174361858629621432, 3.24229584203906335485827551561, 3.69656774805218198534431793330, 4.50426503132686146127872626357, 4.66043629966342784974621871937, 5.57563154157219914651930065485, 6.56845343610710827421950839293, 7.40093200900812343193993508106, 7.42769423515147453862951066724, 7.86935605681606903657724655349, 7.87526213274663625327873522313, 8.639076508177464065000002641327, 8.777510517566518098706293840748, 9.327444753066146962134264057681, 9.578180479679418414146930744942, 10.39823081149817439046632654717, 10.43343526796892694157864761346
