| L(s) = 1 | − 3-s − 3·4-s − 9·7-s + 9-s + 3·12-s − 11·13-s + 5·16-s − 11·19-s + 9·21-s − 6·25-s − 27-s + 27·28-s − 4·31-s − 3·36-s − 17·37-s + 11·39-s − 6·43-s − 5·48-s + 47·49-s + 33·52-s + 11·57-s + 4·61-s − 9·63-s − 3·64-s + 4·67-s − 15·73-s + 6·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 3/2·4-s − 3.40·7-s + 1/3·9-s + 0.866·12-s − 3.05·13-s + 5/4·16-s − 2.52·19-s + 1.96·21-s − 6/5·25-s − 0.192·27-s + 5.10·28-s − 0.718·31-s − 1/2·36-s − 2.79·37-s + 1.76·39-s − 0.914·43-s − 0.721·48-s + 47/7·49-s + 4.57·52-s + 1.45·57-s + 0.512·61-s − 1.13·63-s − 3/8·64-s + 0.488·67-s − 1.75·73-s + 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1069713 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1069713 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 39619 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 240 T + p T^{2} ) \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 11 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 132 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.10531123862267442755720886101, −6.91068861125947648461952084294, −6.74770088061042244190812925387, −6.00415044082805906307705047103, −5.67765143802959162396333307372, −5.22845980030825968092963043495, −4.56988781748648434498307287256, −4.27762540666330579764028769231, −3.64325917342490411095358911798, −3.32944458358972315161071277409, −2.56484736502393400866351676418, −2.07060944105919913272904773593, 0, 0, 0,
2.07060944105919913272904773593, 2.56484736502393400866351676418, 3.32944458358972315161071277409, 3.64325917342490411095358911798, 4.27762540666330579764028769231, 4.56988781748648434498307287256, 5.22845980030825968092963043495, 5.67765143802959162396333307372, 6.00415044082805906307705047103, 6.74770088061042244190812925387, 6.91068861125947648461952084294, 7.10531123862267442755720886101
