| L(s) = 1 | − 2·2-s − 5-s − 3·7-s + 4·8-s − 4·9-s + 2·10-s + 13-s + 6·14-s − 4·16-s − 8·17-s + 8·18-s − 3·19-s + 3·23-s − 4·25-s − 2·26-s + 6·29-s − 3·31-s + 16·34-s + 3·35-s + 4·37-s + 6·38-s − 4·40-s + 4·41-s + 7·43-s + 4·45-s − 6·46-s − 6·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.447·5-s − 1.13·7-s + 1.41·8-s − 4/3·9-s + 0.632·10-s + 0.277·13-s + 1.60·14-s − 16-s − 1.94·17-s + 1.88·18-s − 0.688·19-s + 0.625·23-s − 4/5·25-s − 0.392·26-s + 1.11·29-s − 0.538·31-s + 2.74·34-s + 0.507·35-s + 0.657·37-s + 0.973·38-s − 0.632·40-s + 0.624·41-s + 1.06·43-s + 0.596·45-s − 0.884·46-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3499 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3499 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3499 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 55 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - T + 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 3 T - 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T - 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T - 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 7 T + 28 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T - 8 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 13 T + 115 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 11 T + 148 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 112 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 7 T + 24 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - T + 54 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 92 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 54 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−18.1608582001, −17.7713894249, −17.3269485592, −16.9780551527, −16.3392357664, −15.8421805662, −15.3709389458, −14.6225065853, −14.0030696211, −13.3748806059, −13.0580808016, −12.3244823174, −11.5073139527, −10.9455195866, −10.5444422054, −9.60911625660, −9.22570523603, −8.77350820758, −8.26942226728, −7.61321808621, −6.56461123607, −6.11053388620, −4.83506232904, −3.96729595574, −2.71208851284, 0,
2.71208851284, 3.96729595574, 4.83506232904, 6.11053388620, 6.56461123607, 7.61321808621, 8.26942226728, 8.77350820758, 9.22570523603, 9.60911625660, 10.5444422054, 10.9455195866, 11.5073139527, 12.3244823174, 13.0580808016, 13.3748806059, 14.0030696211, 14.6225065853, 15.3709389458, 15.8421805662, 16.3392357664, 16.9780551527, 17.3269485592, 17.7713894249, 18.1608582001
