| L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s + 2·7-s − 3·8-s + 3·9-s − 10-s − 9·11-s + 2·12-s + 4·13-s − 2·14-s + 2·15-s + 16-s − 4·17-s − 3·18-s − 9·19-s + 20-s + 4·21-s + 9·22-s + 2·23-s − 6·24-s − 5·25-s − 4·26-s + 4·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s − 1.06·8-s + 9-s − 0.316·10-s − 2.71·11-s + 0.577·12-s + 1.10·13-s − 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.970·17-s − 0.707·18-s − 2.06·19-s + 0.223·20-s + 0.872·21-s + 1.91·22-s + 0.417·23-s − 1.22·24-s − 25-s − 0.784·26-s + 0.769·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64689849 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64689849 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 383 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 6 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 9 T + 38 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + 9 T + 54 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 50 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 3 T + 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_4$ | \( 1 - 15 T + 146 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 86 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 98 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 148 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 25 T + 330 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 226 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−7.76737368448892512548724295295, −7.39751357962585742091464981269, −7.21231082826054168397471167754, −6.59389734114462503300429265674, −6.46506587496011153327561573421, −5.84096348457797270911293338167, −5.68309803314568383917737269824, −5.31473957602795572985743482486, −4.79241108969286974165110558026, −4.47702651977205830771714179261, −3.94549744194654548524420319266, −3.73664676378171984159024244614, −3.09147949644978268675440171750, −2.64633050743825331430755292951, −2.37464279368560100017488420719, −2.10801075472669542844270756903, −1.80701446965886015032634784315, −1.06348592998466505165400905169, 0, 0,
1.06348592998466505165400905169, 1.80701446965886015032634784315, 2.10801075472669542844270756903, 2.37464279368560100017488420719, 2.64633050743825331430755292951, 3.09147949644978268675440171750, 3.73664676378171984159024244614, 3.94549744194654548524420319266, 4.47702651977205830771714179261, 4.79241108969286974165110558026, 5.31473957602795572985743482486, 5.68309803314568383917737269824, 5.84096348457797270911293338167, 6.46506587496011153327561573421, 6.59389734114462503300429265674, 7.21231082826054168397471167754, 7.39751357962585742091464981269, 7.76737368448892512548724295295
