| L(s) = 1 | + 2-s + 3-s − 4-s + 2·5-s + 6-s − 8-s − 9-s + 2·10-s + 2·11-s − 12-s − 3·13-s + 2·15-s − 16-s + 17-s − 18-s − 7·19-s − 2·20-s + 2·22-s − 24-s + 3·25-s − 3·26-s + 2·30-s − 31-s − 5·32-s + 2·33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s − 1/3·9-s + 0.632·10-s + 0.603·11-s − 0.288·12-s − 0.832·13-s + 0.516·15-s − 1/4·16-s + 0.242·17-s − 0.235·18-s − 1.60·19-s − 0.447·20-s + 0.426·22-s − 0.204·24-s + 3/5·25-s − 0.588·26-s + 0.365·30-s − 0.179·31-s − 0.883·32-s + 0.348·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10075 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10075 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.535741831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535741831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 2 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T - 16 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 7 T + 42 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 5 T + 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 82 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 89 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−16.5636796811, −16.2045096153, −15.2810900540, −14.7798044476, −14.5899058557, −14.0030225231, −13.8039606277, −13.1516890780, −12.6887432179, −12.3908667253, −11.5781147081, −10.9495831637, −10.3312807952, −9.80107689490, −9.12064955391, −8.85793764388, −8.23203302303, −7.35777663312, −6.71124713767, −5.96198559233, −5.38742169503, −4.53279690025, −4.06627027500, −2.93834588333, −2.04902382484,
2.04902382484, 2.93834588333, 4.06627027500, 4.53279690025, 5.38742169503, 5.96198559233, 6.71124713767, 7.35777663312, 8.23203302303, 8.85793764388, 9.12064955391, 9.80107689490, 10.3312807952, 10.9495831637, 11.5781147081, 12.3908667253, 12.6887432179, 13.1516890780, 13.8039606277, 14.0030225231, 14.5899058557, 14.7798044476, 15.2810900540, 16.2045096153, 16.5636796811
