| L(s) = 1 | − 2·2-s − 2·3-s + 2·5-s + 4·6-s − 2·7-s + 4·8-s − 4·10-s − 4·11-s − 13-s + 4·14-s − 4·15-s − 4·16-s − 6·17-s − 4·19-s + 4·21-s + 8·22-s − 3·23-s − 8·24-s − 25-s + 2·26-s + 2·27-s + 5·29-s + 8·30-s + 5·31-s + 8·33-s + 12·34-s − 4·35-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 0.894·5-s + 1.63·6-s − 0.755·7-s + 1.41·8-s − 1.26·10-s − 1.20·11-s − 0.277·13-s + 1.06·14-s − 1.03·15-s − 16-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 1.70·22-s − 0.625·23-s − 1.63·24-s − 1/5·25-s + 0.392·26-s + 0.384·27-s + 0.928·29-s + 1.46·30-s + 0.898·31-s + 1.39·33-s + 2.05·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3103 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3103 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| 107 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 17 T + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 T + p T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 5 T + 3 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 58 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 80 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 + 10 T + 92 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - T + 74 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 92 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7 T + 189 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
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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
−18.2527534204, −17.7414234899, −17.5554386124, −17.2261220453, −16.6196237274, −16.1400847436, −15.6000401410, −14.9175152572, −13.9334878307, −13.5354717169, −13.0934360577, −12.5000430421, −11.7242855410, −10.9486478024, −10.5297659270, −9.94093551657, −9.62071646854, −8.80026448571, −8.35579926119, −7.56038144260, −6.43222986470, −6.11142068177, −5.14312914143, −4.40457014342, −2.47137165063, 0,
2.47137165063, 4.40457014342, 5.14312914143, 6.11142068177, 6.43222986470, 7.56038144260, 8.35579926119, 8.80026448571, 9.62071646854, 9.94093551657, 10.5297659270, 10.9486478024, 11.7242855410, 12.5000430421, 13.0934360577, 13.5354717169, 13.9334878307, 14.9175152572, 15.6000401410, 16.1400847436, 16.6196237274, 17.2261220453, 17.5554386124, 17.7414234899, 18.2527534204
