| L(s) = 1 | − 3-s − 5-s − 5·7-s + 11-s + 15-s + 8·17-s − 4·19-s + 5·21-s + 4·23-s + 5·25-s + 27-s − 10·29-s + 7·31-s − 33-s + 5·35-s − 8·37-s + 8·41-s − 20·43-s + 6·47-s + 18·49-s − 8·51-s + 53-s − 55-s + 4·57-s + 9·59-s + 2·61-s + 2·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.88·7-s + 0.301·11-s + 0.258·15-s + 1.94·17-s − 0.917·19-s + 1.09·21-s + 0.834·23-s + 25-s + 0.192·27-s − 1.85·29-s + 1.25·31-s − 0.174·33-s + 0.845·35-s − 1.31·37-s + 1.24·41-s − 3.04·43-s + 0.875·47-s + 18/7·49-s − 1.12·51-s + 0.137·53-s − 0.134·55-s + 0.529·57-s + 1.17·59-s + 0.256·61-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7558208049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7558208049\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - T - 52 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 9 T + 22 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−10.38335231011406337288309760933, −10.24232467383505110227878014046, −10.18919589710019842251421291485, −9.417290954264497501034367123020, −8.914476814037204582368804569796, −8.854180000831933587142880249798, −8.000432941786929603211890496709, −7.66298732854043910404128063322, −6.99411027069282291865501315645, −6.72270821779458452960287528656, −6.40703718155769389690382344059, −5.69871967110398399890935357189, −5.44810661606830710339852772417, −4.89048151886978829215894601959, −3.99215098673541023190041826701, −3.71176024187179035900510837546, −3.10565073484263366235437577474, −2.70077981767122193049153429156, −1.48526908594197870739082815684, −0.50101183626140419904358975111,
0.50101183626140419904358975111, 1.48526908594197870739082815684, 2.70077981767122193049153429156, 3.10565073484263366235437577474, 3.71176024187179035900510837546, 3.99215098673541023190041826701, 4.89048151886978829215894601959, 5.44810661606830710339852772417, 5.69871967110398399890935357189, 6.40703718155769389690382344059, 6.72270821779458452960287528656, 6.99411027069282291865501315645, 7.66298732854043910404128063322, 8.000432941786929603211890496709, 8.854180000831933587142880249798, 8.914476814037204582368804569796, 9.417290954264497501034367123020, 10.18919589710019842251421291485, 10.24232467383505110227878014046, 10.38335231011406337288309760933
