| L(s) = 1 | − 4·3-s − 4·5-s − 7-s + 7·9-s − 2·11-s + 2·13-s + 16·15-s + 17-s − 2·19-s + 4·21-s − 23-s + 6·25-s − 4·27-s − 6·29-s − 5·31-s + 8·33-s + 4·35-s − 2·37-s − 8·39-s − 12·41-s + 6·43-s − 28·45-s + 3·47-s + 9·49-s − 4·51-s + 8·55-s + 8·57-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.78·5-s − 0.377·7-s + 7/3·9-s − 0.603·11-s + 0.554·13-s + 4.13·15-s + 0.242·17-s − 0.458·19-s + 0.872·21-s − 0.208·23-s + 6/5·25-s − 0.769·27-s − 1.11·29-s − 0.898·31-s + 1.39·33-s + 0.676·35-s − 0.328·37-s − 1.28·39-s − 1.87·41-s + 0.914·43-s − 4.17·45-s + 0.437·47-s + 9/7·49-s − 0.560·51-s + 1.07·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2336 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 7 T + p T^{2} ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + T - 8 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T - 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + T - 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 36 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 + 2 T + 9 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 12 T + 113 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 28 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 68 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 111 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 134 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 52 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 154 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 70 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−18.7910727934, −18.3101687612, −17.8259013675, −17.0370558147, −16.8185078546, −16.2972040061, −15.8431901143, −15.3902728740, −14.9205895459, −13.8798699581, −13.0840932222, −12.4813953098, −12.0128484984, −11.6801874308, −10.9752876462, −10.8445854818, −10.1051784220, −8.98347607555, −8.16611554241, −7.40917093916, −6.80809096343, −5.87118938712, −5.42327471098, −4.42561115543, −3.57790890049, 0,
3.57790890049, 4.42561115543, 5.42327471098, 5.87118938712, 6.80809096343, 7.40917093916, 8.16611554241, 8.98347607555, 10.1051784220, 10.8445854818, 10.9752876462, 11.6801874308, 12.0128484984, 12.4813953098, 13.0840932222, 13.8798699581, 14.9205895459, 15.3902728740, 15.8431901143, 16.2972040061, 16.8185078546, 17.0370558147, 17.8259013675, 18.3101687612, 18.7910727934
