| L(s) = 1 | + 3-s − 4-s − 2·5-s − 2·11-s − 12-s − 13-s − 2·15-s − 17-s + 2·20-s + 3·25-s − 27-s − 2·29-s − 2·33-s − 39-s + 2·44-s − 47-s − 51-s + 52-s + 4·55-s + 2·60-s + 64-s + 2·65-s + 68-s − 2·71-s − 73-s + 3·75-s + 79-s + ⋯ |
| L(s) = 1 | + 3-s − 4-s − 2·5-s − 2·11-s − 12-s − 13-s − 2·15-s − 17-s + 2·20-s + 3·25-s − 27-s − 2·29-s − 2·33-s − 39-s + 2·44-s − 47-s − 51-s + 52-s + 4·55-s + 2·60-s + 64-s + 2·65-s + 68-s − 2·71-s − 73-s + 3·75-s + 79-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
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02329293517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02329293517\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 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
−9.566671981594095622006622633838, −8.859568537909559257021859357092, −8.637373265685509589425506990638, −8.238585446417142323135468133399, −7.974274197515447138288387367757, −7.62313981504224925251360684452, −7.32035518829285188418376571392, −7.06772490830395878993606070776, −6.44522424421298641649971969984, −5.68998487486169094848447714222, −5.27048683863024817261348554411, −4.93516210497033310093924648800, −4.58087534062662674520746157327, −3.97150994849339384102716452276, −3.92035110448656786628469104350, −3.21809515910575157188508358647, −2.64682065646636589172963596635, −2.60676946862025879113476950774, −1.65548428536180489242955050933, −0.095927526189540525474787946843,
0.095927526189540525474787946843, 1.65548428536180489242955050933, 2.60676946862025879113476950774, 2.64682065646636589172963596635, 3.21809515910575157188508358647, 3.92035110448656786628469104350, 3.97150994849339384102716452276, 4.58087534062662674520746157327, 4.93516210497033310093924648800, 5.27048683863024817261348554411, 5.68998487486169094848447714222, 6.44522424421298641649971969984, 7.06772490830395878993606070776, 7.32035518829285188418376571392, 7.62313981504224925251360684452, 7.974274197515447138288387367757, 8.238585446417142323135468133399, 8.637373265685509589425506990638, 8.859568537909559257021859357092, 9.566671981594095622006622633838
