| L(s) = 1 | − 2-s − 3-s − 2·4-s + 6-s − 2·7-s + 3·8-s − 2·9-s − 11-s + 2·12-s − 2·13-s + 2·14-s + 16-s − 8·17-s + 2·18-s − 11·19-s + 2·21-s + 22-s + 2·23-s − 3·24-s − 4·25-s + 2·26-s + 2·27-s + 4·28-s − 2·29-s − 8·31-s − 2·32-s + 33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 4-s + 0.408·6-s − 0.755·7-s + 1.06·8-s − 2/3·9-s − 0.301·11-s + 0.577·12-s − 0.554·13-s + 0.534·14-s + 1/4·16-s − 1.94·17-s + 0.471·18-s − 2.52·19-s + 0.436·21-s + 0.213·22-s + 0.417·23-s − 0.612·24-s − 4/5·25-s + 0.392·26-s + 0.384·27-s + 0.755·28-s − 0.371·29-s − 1.43·31-s − 0.353·32-s + 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100315 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100315 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| 20063 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 88 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T + 7 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 11 T + 67 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 60 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 29 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T - 36 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 82 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 15 T + 133 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 64 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 147 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 75 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 113 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3 T + 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 3 T + 84 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−14.5260485149, −13.9380336131, −13.4358926221, −13.1708140849, −12.8436846972, −12.4358467828, −11.7894651162, −11.1712945783, −10.9902331263, −10.3604382215, −10.1254703921, −9.32067308380, −9.12577062292, −8.66920298751, −8.40665720869, −7.63920583757, −7.12567479615, −6.33467636650, −6.23040956640, −5.40592570469, −4.84288353546, −4.23081342044, −3.85842009506, −2.67467518487, −2.02034668018, 0, 0,
2.02034668018, 2.67467518487, 3.85842009506, 4.23081342044, 4.84288353546, 5.40592570469, 6.23040956640, 6.33467636650, 7.12567479615, 7.63920583757, 8.40665720869, 8.66920298751, 9.12577062292, 9.32067308380, 10.1254703921, 10.3604382215, 10.9902331263, 11.1712945783, 11.7894651162, 12.4358467828, 12.8436846972, 13.1708140849, 13.4358926221, 13.9380336131, 14.5260485149
