| L(s) = 1 | + 2-s + 4·7-s + 8-s − 5·9-s − 2·11-s + 6·13-s + 4·14-s − 16-s + 3·17-s − 5·18-s − 2·19-s − 2·22-s + 2·23-s + 2·25-s + 6·26-s − 9·29-s + 8·31-s − 6·32-s + 3·34-s + 7·37-s − 2·38-s − 6·41-s − 8·43-s + 2·46-s − 14·47-s + 7·49-s + 2·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·7-s + 0.353·8-s − 5/3·9-s − 0.603·11-s + 1.66·13-s + 1.06·14-s − 1/4·16-s + 0.727·17-s − 1.17·18-s − 0.458·19-s − 0.426·22-s + 0.417·23-s + 2/5·25-s + 1.17·26-s − 1.67·29-s + 1.43·31-s − 1.06·32-s + 0.514·34-s + 1.15·37-s − 0.324·38-s − 0.937·41-s − 1.21·43-s + 0.294·46-s − 2.04·47-s + 49-s + 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17593 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17593 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.643788275\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643788275\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 11 T + p T^{2} ) \) |
| 241 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 34 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 130 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 13 T + 112 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T - 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 141 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 104 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 23 T + 292 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 30 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−15.7849704026, −15.1992848657, −14.7865123446, −14.5075214574, −13.9058703620, −13.6096220164, −13.1701562714, −12.7129548859, −11.7302798830, −11.4700638843, −11.1979682155, −10.7267223224, −10.0783879428, −9.20178836683, −8.55138194998, −8.23491999302, −7.93193688803, −6.95908163603, −6.21653069534, −5.50054922879, −5.19354714673, −4.47324500652, −3.66309834968, −2.82333387766, −1.63153449557,
1.63153449557, 2.82333387766, 3.66309834968, 4.47324500652, 5.19354714673, 5.50054922879, 6.21653069534, 6.95908163603, 7.93193688803, 8.23491999302, 8.55138194998, 9.20178836683, 10.0783879428, 10.7267223224, 11.1979682155, 11.4700638843, 11.7302798830, 12.7129548859, 13.1701562714, 13.6096220164, 13.9058703620, 14.5075214574, 14.7865123446, 15.1992848657, 15.7849704026
