| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s − 5·5-s + 6·6-s − 2·7-s − 3·8-s + 15·10-s − 2·11-s − 8·12-s − 8·13-s + 6·14-s + 10·15-s + 3·16-s − 3·17-s + 2·19-s − 20·20-s + 4·21-s + 6·22-s + 6·24-s + 12·25-s + 24·26-s + 2·27-s − 8·28-s − 10·29-s − 30·30-s + 6·31-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 + 4.74·10-s − 0.603·11-s − 2.30·12-s − 2.21·13-s + 1.60·14-s + 2.58·15-s + 3/4·16-s − 0.727·17-s + 0.458·19-s − 4.47·20-s + 0.872·21-s + 1.27·22-s + 1.22·24-s + 12/5·25-s + 4.70·26-s + 0.384·27-s − 1.51·28-s − 1.85·29-s − 5.47·30-s + 1.07·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7609 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 3 T + p T^{2} ) \) |
| 1087 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 4 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p T + 13 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 8 T + 34 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 10 T + 53 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 56 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 48 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 9 T + 19 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 164 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 180 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−17.3360338471, −17.0789596554, −16.7608800502, −16.0882490632, −15.7514865002, −15.2756499269, −14.9089982866, −14.0448424271, −13.0672755219, −12.3220946485, −12.1556382181, −11.6317147129, −11.1241215527, −10.6536479577, −9.98913471408, −9.56313053645, −8.89499775694, −8.31092614900, −7.62350012324, −7.41807613300, −6.84157717295, −5.70387683136, −4.94127209794, −3.99268703019, −2.83342087119, 0, 0,
2.83342087119, 3.99268703019, 4.94127209794, 5.70387683136, 6.84157717295, 7.41807613300, 7.62350012324, 8.31092614900, 8.89499775694, 9.56313053645, 9.98913471408, 10.6536479577, 11.1241215527, 11.6317147129, 12.1556382181, 12.3220946485, 13.0672755219, 14.0448424271, 14.9089982866, 15.2756499269, 15.7514865002, 16.0882490632, 16.7608800502, 17.0789596554, 17.3360338471
