| L(s) = 1 | + 3·3-s − 4·5-s + 2·7-s + 6·9-s + 5·11-s + 2·13-s − 12·15-s − 6·17-s + 2·19-s + 6·21-s + 6·23-s + 5·25-s + 9·27-s + 2·29-s + 4·31-s + 15·33-s − 8·35-s − 16·37-s + 6·39-s − 41-s + 7·43-s − 24·45-s − 2·47-s + 7·49-s − 18·51-s − 8·53-s − 20·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s + 0.755·7-s + 2·9-s + 1.50·11-s + 0.554·13-s − 3.09·15-s − 1.45·17-s + 0.458·19-s + 1.30·21-s + 1.25·23-s + 25-s + 1.73·27-s + 0.371·29-s + 0.718·31-s + 2.61·33-s − 1.35·35-s − 2.63·37-s + 0.960·39-s − 0.156·41-s + 1.06·43-s − 3.57·45-s − 0.291·47-s + 49-s − 2.52·51-s − 1.09·53-s − 2.69·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.398292271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.398292271\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + T - 40 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 7 T + 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 11 T + 24 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−11.96515032737071359011476631641, −11.58992605102521580414591388366, −11.06884016155214542173628724788, −10.93052300879172850164363462611, −10.04405284622934810517354496189, −9.462870364298880640375908613658, −8.907115522486334528310437285625, −8.725221682459801454803170197823, −8.241042936254376101593367325963, −7.958206778465429931682628098729, −7.10706274055284524913545242576, −7.06665273115700331922668475132, −6.45918893759429890462372505369, −5.29320261988475748814649362436, −4.42918310675808387557640516636, −4.26720227570453963440074551077, −3.56234084650169870004559774881, −3.24151522836286798397862704477, −2.18548565240196677980646464441, −1.26250738470743077404624406683,
1.26250738470743077404624406683, 2.18548565240196677980646464441, 3.24151522836286798397862704477, 3.56234084650169870004559774881, 4.26720227570453963440074551077, 4.42918310675808387557640516636, 5.29320261988475748814649362436, 6.45918893759429890462372505369, 7.06665273115700331922668475132, 7.10706274055284524913545242576, 7.958206778465429931682628098729, 8.241042936254376101593367325963, 8.725221682459801454803170197823, 8.907115522486334528310437285625, 9.462870364298880640375908613658, 10.04405284622934810517354496189, 10.93052300879172850164363462611, 11.06884016155214542173628724788, 11.58992605102521580414591388366, 11.96515032737071359011476631641
