| L(s) = 1 | + 2-s − 3-s + 2·5-s − 6-s − 2·7-s + 8-s + 2·10-s + 5·11-s − 3·13-s − 2·14-s − 2·15-s − 16-s + 2·17-s − 2·19-s + 2·21-s + 5·22-s − 24-s − 3·25-s − 3·26-s + 4·27-s + 3·29-s − 2·30-s − 3·31-s − 6·32-s − 5·33-s + 2·34-s − 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.894·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 0.632·10-s + 1.50·11-s − 0.832·13-s − 0.534·14-s − 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.458·19-s + 0.436·21-s + 1.06·22-s − 0.204·24-s − 3/5·25-s − 0.588·26-s + 0.769·27-s + 0.557·29-s − 0.365·30-s − 0.538·31-s − 1.06·32-s − 0.870·33-s + 0.342·34-s − 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7161 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7161 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.139759179\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.139759179\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 - T )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 6 T + p T^{2} ) \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 6 T - 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 118 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + T + 32 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T - 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−16.9457067265, −16.5231813814, −16.2667339668, −15.4630072916, −14.8373325679, −14.3572818594, −13.9919753005, −13.4895168657, −12.9806533587, −12.4584414597, −11.8813276077, −11.5796849843, −10.6635245733, −10.1560406464, −9.69960549853, −9.07643597137, −8.53350919251, −7.34518516465, −6.85264890082, −6.23039069399, −5.67835761722, −4.89508231922, −4.20044556433, −3.27957952917, −1.89667356298,
1.89667356298, 3.27957952917, 4.20044556433, 4.89508231922, 5.67835761722, 6.23039069399, 6.85264890082, 7.34518516465, 8.53350919251, 9.07643597137, 9.69960549853, 10.1560406464, 10.6635245733, 11.5796849843, 11.8813276077, 12.4584414597, 12.9806533587, 13.4895168657, 13.9919753005, 14.3572818594, 14.8373325679, 15.4630072916, 16.2667339668, 16.5231813814, 16.9457067265
