| L(s) = 1 | + 2-s − 3·3-s + 4-s − 2·5-s − 3·6-s + 8-s + 3·9-s − 2·10-s + 11-s − 3·12-s + 6·15-s + 16-s − 5·17-s + 3·18-s − 5·19-s − 2·20-s + 22-s − 4·23-s − 3·24-s − 25-s + 4·29-s + 6·30-s − 8·31-s + 32-s − 3·33-s − 5·34-s + 3·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s − 1.22·6-s + 0.353·8-s + 9-s − 0.632·10-s + 0.301·11-s − 0.866·12-s + 1.54·15-s + 1/4·16-s − 1.21·17-s + 0.707·18-s − 1.14·19-s − 0.447·20-s + 0.213·22-s − 0.834·23-s − 0.612·24-s − 1/5·25-s + 0.742·29-s + 1.09·30-s − 1.43·31-s + 0.176·32-s − 0.522·33-s − 0.857·34-s + 1/2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 - T \) |
| 5 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 30 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T - 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 62 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 34 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 62 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 34 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 30 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 2 T - 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - T + 88 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 14 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 17 T + 174 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 11 T + 92 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−15.8559089386, −15.8202897091, −15.0566589154, −14.7163121675, −14.1605238137, −13.4415825519, −13.0527470601, −12.4794639226, −12.0286686569, −11.6963533906, −11.2641621174, −10.8769489628, −10.4911486048, −9.75905593192, −8.92857496325, −8.35318486363, −7.71194526503, −6.90781892989, −6.56628544466, −5.91251630833, −5.50252070738, −4.55718085134, −4.30434203350, −3.40737633752, −2.05023038526, 0,
2.05023038526, 3.40737633752, 4.30434203350, 4.55718085134, 5.50252070738, 5.91251630833, 6.56628544466, 6.90781892989, 7.71194526503, 8.35318486363, 8.92857496325, 9.75905593192, 10.4911486048, 10.8769489628, 11.2641621174, 11.6963533906, 12.0286686569, 12.4794639226, 13.0527470601, 13.4415825519, 14.1605238137, 14.7163121675, 15.0566589154, 15.8202897091, 15.8559089386
