| L(s) = 1 | − 2-s + 3-s − 2·4-s − 5-s − 6-s + 3·8-s + 9-s + 10-s − 6·11-s − 2·12-s + 13-s − 15-s + 16-s + 4·17-s − 18-s + 2·20-s + 6·22-s + 6·23-s + 3·24-s − 4·25-s − 26-s + 27-s + 2·29-s + 30-s − 4·31-s − 2·32-s − 6·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 4-s − 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s − 0.577·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.447·20-s + 1.27·22-s + 1.25·23-s + 0.612·24-s − 4/5·25-s − 0.196·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s − 0.718·31-s − 0.353·32-s − 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100035 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100035 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + p T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + p T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 24 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 - T - 74 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 3 T - 46 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 105 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 17 T + 196 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 78 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 5 T + 178 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
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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
−14.3182182802, −13.6791203364, −13.3562933455, −13.1847398160, −12.6053560437, −12.1753106139, −11.6799676714, −10.7792657833, −10.7171639889, −10.1965884440, −9.73707630115, −9.14610703071, −8.89231924120, −8.44426556460, −7.84676406921, −7.45176369658, −7.40493950553, −6.20202908477, −5.63622245063, −5.02699315989, −4.58887381257, −3.89034771754, −3.18386250650, −2.58659110698, −1.31411593906, 0,
1.31411593906, 2.58659110698, 3.18386250650, 3.89034771754, 4.58887381257, 5.02699315989, 5.63622245063, 6.20202908477, 7.40493950553, 7.45176369658, 7.84676406921, 8.44426556460, 8.89231924120, 9.14610703071, 9.73707630115, 10.1965884440, 10.7171639889, 10.7792657833, 11.6799676714, 12.1753106139, 12.6053560437, 13.1847398160, 13.3562933455, 13.6791203364, 14.3182182802
