| L(s) = 1 | − 2-s − 3-s − 5-s + 6-s − 7-s + 8-s + 10-s − 3·11-s − 4·13-s + 14-s + 15-s − 16-s + 17-s − 3·19-s + 21-s + 3·22-s − 3·23-s − 24-s − 4·25-s + 4·26-s + 4·27-s − 2·29-s − 30-s + 31-s + 3·33-s − 34-s + 35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 0.904·11-s − 1.10·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.242·17-s − 0.688·19-s + 0.218·21-s + 0.639·22-s − 0.625·23-s − 0.204·24-s − 4/5·25-s + 0.784·26-s + 0.769·27-s − 0.371·29-s − 0.182·30-s + 0.179·31-s + 0.522·33-s − 0.171·34-s + 0.169·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10020 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10020 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 167 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 18 T + p T^{2} ) \) |
| good | 7 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 17 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 2 T + 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 45 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 10 T + 88 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 72 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $D_{4}$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 67 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 65 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T - 8 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 9 T + 91 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 19 T + 243 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T + 121 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−16.9772999466, −16.4584351567, −15.9257490830, −15.4264689887, −15.1634546807, −14.3045354452, −13.9143631533, −13.2977612682, −12.6270507967, −12.2704988135, −11.8374617096, −11.1115705847, −10.5612333389, −10.1990709523, −9.61861897387, −9.07532049080, −8.16422827496, −7.98092744902, −7.19079527655, −6.62804183746, −5.77078173235, −5.10275854314, −4.40572435272, −3.37269685281, −2.19283119970, 0,
2.19283119970, 3.37269685281, 4.40572435272, 5.10275854314, 5.77078173235, 6.62804183746, 7.19079527655, 7.98092744902, 8.16422827496, 9.07532049080, 9.61861897387, 10.1990709523, 10.5612333389, 11.1115705847, 11.8374617096, 12.2704988135, 12.6270507967, 13.2977612682, 13.9143631533, 14.3045354452, 15.1634546807, 15.4264689887, 15.9257490830, 16.4584351567, 16.9772999466
