| L(s) = 1 | + 2·3-s + 4-s + 6·5-s + 7-s + 2·9-s − 6·11-s + 2·12-s + 2·13-s + 12·15-s − 3·16-s + 2·17-s + 13·19-s + 6·20-s + 2·21-s − 8·23-s + 17·25-s + 6·27-s + 28-s − 3·29-s + 31-s − 12·33-s + 6·35-s + 2·36-s − 6·37-s + 4·39-s − 41-s + 6·43-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s + 2.68·5-s + 0.377·7-s + 2/3·9-s − 1.80·11-s + 0.577·12-s + 0.554·13-s + 3.09·15-s − 3/4·16-s + 0.485·17-s + 2.98·19-s + 1.34·20-s + 0.436·21-s − 1.66·23-s + 17/5·25-s + 1.15·27-s + 0.188·28-s − 0.557·29-s + 0.179·31-s − 2.08·33-s + 1.01·35-s + 1/3·36-s − 0.986·37-s + 0.640·39-s − 0.156·41-s + 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.479648731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.479648731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 109 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 563 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 23 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 54 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T - 54 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T - 14 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + T + 106 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T - 2 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 18 T + 175 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 21 T + 240 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
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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
−12.9335173631, −12.5871168009, −11.9293223064, −11.6392512421, −10.9544427746, −10.5107720757, −10.1579134002, −9.88107504710, −9.56811281828, −9.07833458795, −8.78147499895, −8.03660422819, −7.79410771075, −7.33699740070, −6.79930324211, −6.14771963537, −5.68332539040, −5.46049938972, −5.06880539395, −4.27489338613, −3.22479125780, −2.89255161087, −2.46265088880, −1.78574662236, −1.43693342297,
1.43693342297, 1.78574662236, 2.46265088880, 2.89255161087, 3.22479125780, 4.27489338613, 5.06880539395, 5.46049938972, 5.68332539040, 6.14771963537, 6.79930324211, 7.33699740070, 7.79410771075, 8.03660422819, 8.78147499895, 9.07833458795, 9.56811281828, 9.88107504710, 10.1579134002, 10.5107720757, 10.9544427746, 11.6392512421, 11.9293223064, 12.5871168009, 12.9335173631
