| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 5·7-s − 3·8-s − 2·9-s + 11-s + 2·12-s − 6·13-s + 5·14-s + 16-s − 3·17-s + 2·18-s − 4·19-s − 10·21-s − 22-s − 6·24-s − 6·25-s + 6·26-s − 10·27-s − 5·28-s − 2·29-s + 3·31-s + 32-s + 2·33-s + 3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.88·7-s − 1.06·8-s − 2/3·9-s + 0.301·11-s + 0.577·12-s − 1.66·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s + 0.471·18-s − 0.917·19-s − 2.18·21-s − 0.213·22-s − 1.22·24-s − 6/5·25-s + 1.17·26-s − 1.92·27-s − 0.944·28-s − 0.371·29-s + 0.538·31-s + 0.176·32-s + 0.348·33-s + 0.514·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 610423 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 610423 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 211 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - T + p T^{2} ) \) |
| 263 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 18 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 25 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 2 T - 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 3 T + 42 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9 T + 51 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 13 T + 92 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 92 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 79 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T - 10 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 35 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 36 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−12.9858166912, −12.3857934759, −12.1508714005, −11.6950635736, −11.3674505942, −10.8476594999, −10.1347554111, −9.94791340487, −9.47864565123, −9.28872576854, −8.94438199991, −8.51778912630, −7.99401519115, −7.72957056862, −7.03259155712, −6.56150091501, −6.37839530471, −5.87256700021, −5.28177033240, −4.48163961542, −3.87516942456, −3.30214676964, −2.75367433548, −2.57980782423, −1.94990056431, 0, 0,
1.94990056431, 2.57980782423, 2.75367433548, 3.30214676964, 3.87516942456, 4.48163961542, 5.28177033240, 5.87256700021, 6.37839530471, 6.56150091501, 7.03259155712, 7.72957056862, 7.99401519115, 8.51778912630, 8.94438199991, 9.28872576854, 9.47864565123, 9.94791340487, 10.1347554111, 10.8476594999, 11.3674505942, 11.6950635736, 12.1508714005, 12.3857934759, 12.9858166912
