| L(s) = 1 | + 3-s + 4-s − 2·5-s + 12-s − 3·13-s − 2·15-s + 17-s − 2·20-s + 3·25-s − 27-s − 3·39-s + 47-s + 51-s − 3·52-s − 2·60-s − 64-s + 6·65-s + 68-s + 3·73-s + 3·75-s − 79-s − 81-s + 83-s − 2·85-s + 3·97-s + 3·100-s − 108-s + ⋯ |
| L(s) = 1 | + 3-s + 4-s − 2·5-s + 12-s − 3·13-s − 2·15-s + 17-s − 2·20-s + 3·25-s − 27-s − 3·39-s + 47-s + 51-s − 3·52-s − 2·60-s − 64-s + 6·65-s + 68-s + 3·73-s + 3·75-s − 79-s − 81-s + 83-s − 2·85-s + 3·97-s + 3·100-s − 108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.050979091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.050979091\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
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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
−9.600892642100508435393616108660, −8.894319468382598474213094982222, −8.582764038297649325515361448529, −8.181375333548074172316527127893, −7.69276900333386476894245383641, −7.48547974340594690698520736418, −7.36557614668499570565895469317, −7.07974904201735100154039819914, −6.44125132666940221281959470532, −6.07431956913791977435424424864, −5.15566811704490493899886723474, −5.05076801859648638703039400063, −4.65446611592523400748536399203, −3.94270764928727950883937312902, −3.68695707159716777513657252409, −3.12864070778596780235475223129, −2.66663541707616666672849942209, −2.46262747250754668426227903794, −1.84731837072163120855307778092, −0.63544797899156085992677169393,
0.63544797899156085992677169393, 1.84731837072163120855307778092, 2.46262747250754668426227903794, 2.66663541707616666672849942209, 3.12864070778596780235475223129, 3.68695707159716777513657252409, 3.94270764928727950883937312902, 4.65446611592523400748536399203, 5.05076801859648638703039400063, 5.15566811704490493899886723474, 6.07431956913791977435424424864, 6.44125132666940221281959470532, 7.07974904201735100154039819914, 7.36557614668499570565895469317, 7.48547974340594690698520736418, 7.69276900333386476894245383641, 8.181375333548074172316527127893, 8.582764038297649325515361448529, 8.894319468382598474213094982222, 9.600892642100508435393616108660
