| L(s) = 1 | + 3-s + 4-s − 8·7-s + 9-s + 12-s + 4·13-s + 16-s − 8·19-s − 8·21-s + 25-s + 27-s − 8·28-s + 16·31-s + 36-s + 4·37-s + 4·39-s − 8·43-s + 48-s + 34·49-s + 4·52-s − 8·57-s − 20·61-s − 8·63-s + 64-s − 8·67-s + 4·73-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s − 3.02·7-s + 1/3·9-s + 0.288·12-s + 1.10·13-s + 1/4·16-s − 1.83·19-s − 1.74·21-s + 1/5·25-s + 0.192·27-s − 1.51·28-s + 2.87·31-s + 1/6·36-s + 0.657·37-s + 0.640·39-s − 1.21·43-s + 0.144·48-s + 34/7·49-s + 0.554·52-s − 1.05·57-s − 2.56·61-s − 1.00·63-s + 1/8·64-s − 0.977·67-s + 0.468·73-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7472587609\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7472587609\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−13.19720570989419308005927819605, −12.30986793353697911256504493943, −12.14360527021257208948458691105, −10.91913531404329383721387224500, −10.39160509435207624542212960871, −9.880541566397139490214892173475, −9.305587122869086271308905422277, −8.644229019078172602684577487278, −7.941231322257043910223245152113, −6.73106470746564203527567325446, −6.42217617652666865799421983967, −6.11397905422548756742635703073, −4.32687879913691587957737883725, −3.36585804145949552210873743484, −2.74628747875934588873509462774,
2.74628747875934588873509462774, 3.36585804145949552210873743484, 4.32687879913691587957737883725, 6.11397905422548756742635703073, 6.42217617652666865799421983967, 6.73106470746564203527567325446, 7.941231322257043910223245152113, 8.644229019078172602684577487278, 9.305587122869086271308905422277, 9.880541566397139490214892173475, 10.39160509435207624542212960871, 10.91913531404329383721387224500, 12.14360527021257208948458691105, 12.30986793353697911256504493943, 13.19720570989419308005927819605
