| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 6·5-s + 9·6-s − 6·7-s − 3·8-s + 4·9-s + 18·10-s − 5·11-s − 12·12-s − 4·13-s + 18·14-s + 18·15-s + 3·16-s − 7·17-s − 12·18-s − 3·19-s − 24·20-s + 18·21-s + 15·22-s − 2·23-s + 9·24-s + 19·25-s + 12·26-s − 6·27-s − 24·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 2.68·5-s + 3.67·6-s − 2.26·7-s − 1.06·8-s + 4/3·9-s + 5.69·10-s − 1.50·11-s − 3.46·12-s − 1.10·13-s + 4.81·14-s + 4.64·15-s + 3/4·16-s − 1.69·17-s − 2.82·18-s − 0.688·19-s − 5.36·20-s + 3.92·21-s + 3.19·22-s − 0.417·23-s + 1.83·24-s + 19/5·25-s + 2.35·26-s − 1.15·27-s − 4.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39017 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39017 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 4 T + p T^{2} ) \) |
| 3547 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 13 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 7 T + 36 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 7 T + 68 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - T + 48 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 129 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 84 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 14 T + 125 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T - 43 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 169 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 130 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 16 T + 227 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T - 18 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−15.8804402711, −15.7346993436, −15.2036094387, −14.9040135371, −13.5824032771, −13.0107759881, −12.5379766853, −12.3558628818, −11.8641608351, −11.2364718336, −10.9291370967, −10.6411465008, −9.94055353158, −9.65859081140, −9.08662401299, −8.31035345039, −8.07819094418, −7.47613873985, −7.01871973408, −6.55150589611, −5.97043536530, −4.97570788921, −4.38948262978, −3.54881048878, −2.74432662714, 0, 0, 0,
2.74432662714, 3.54881048878, 4.38948262978, 4.97570788921, 5.97043536530, 6.55150589611, 7.01871973408, 7.47613873985, 8.07819094418, 8.31035345039, 9.08662401299, 9.65859081140, 9.94055353158, 10.6411465008, 10.9291370967, 11.2364718336, 11.8641608351, 12.3558628818, 12.5379766853, 13.0107759881, 13.5824032771, 14.9040135371, 15.2036094387, 15.7346993436, 15.8804402711
