| L(s) = 1 | + 2-s + 3-s − 5-s + 6-s − 4·7-s − 8-s − 10-s + 2·11-s + 2·13-s − 4·14-s − 15-s − 16-s − 17-s − 4·19-s − 4·21-s + 2·22-s − 8·23-s − 24-s + 5·25-s + 2·26-s − 27-s + 6·29-s − 30-s + 3·31-s + 2·33-s − 34-s + 4·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 0.447·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.603·11-s + 0.554·13-s − 1.06·14-s − 0.258·15-s − 1/4·16-s − 0.242·17-s − 0.917·19-s − 0.872·21-s + 0.426·22-s − 1.66·23-s − 0.204·24-s + 25-s + 0.392·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.538·31-s + 0.348·33-s − 0.171·34-s + 0.676·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 509796 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 509796 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911876132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911876132\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 8 T + 41 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 11 T + 62 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.47946905391854915008941345020, −10.21110771005934492187322117628, −9.857310598643756610101947839672, −9.138443287715048141997337253603, −9.050796845443754152158729833856, −8.450804944623367116934427594453, −8.223840495622797846490677203246, −7.58554537764802572585635106350, −6.98517740600275583857247075300, −6.54134813905353322593771430461, −6.17020652367192246136571630122, −6.01606544434852511983950118310, −5.04263165218809237201221764705, −4.64721617978732320266774091698, −3.84561735311673590311663966993, −3.82326114511984173412715042913, −3.17758821435670594168676653381, −2.63698772945831770356222991004, −1.91815659072339918073338665907, −0.61377756505634290878075048409,
0.61377756505634290878075048409, 1.91815659072339918073338665907, 2.63698772945831770356222991004, 3.17758821435670594168676653381, 3.82326114511984173412715042913, 3.84561735311673590311663966993, 4.64721617978732320266774091698, 5.04263165218809237201221764705, 6.01606544434852511983950118310, 6.17020652367192246136571630122, 6.54134813905353322593771430461, 6.98517740600275583857247075300, 7.58554537764802572585635106350, 8.223840495622797846490677203246, 8.450804944623367116934427594453, 9.050796845443754152158729833856, 9.138443287715048141997337253603, 9.857310598643756610101947839672, 10.21110771005934492187322117628, 10.47946905391854915008941345020
