| L(s) = 1 | − 2·3-s + 3·9-s − 4·11-s + 2·17-s − 4·19-s + 2·25-s − 4·27-s + 8·33-s + 6·49-s − 4·51-s + 8·57-s + 12·59-s − 20·67-s − 8·73-s − 4·75-s + 5·81-s − 12·83-s − 4·89-s + 20·97-s − 12·99-s − 4·107-s + 4·113-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.20·11-s + 0.485·17-s − 0.917·19-s + 2/5·25-s − 0.769·27-s + 1.39·33-s + 6/7·49-s − 0.560·51-s + 1.05·57-s + 1.56·59-s − 2.44·67-s − 0.936·73-s − 0.461·75-s + 5/9·81-s − 1.31·83-s − 0.423·89-s + 2.03·97-s − 1.20·99-s − 0.386·107-s + 0.376·113-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7842135666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7842135666\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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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
−7.76038911057160832928888509258, −7.55797087757426684077385479442, −7.01612478094549100794464728288, −6.72701301081184558305297995727, −6.02365212856433686585916068303, −5.81185938363869076540861471187, −5.42789212116951928797768738883, −4.86295202326065104196864761496, −4.51050102393655717636124179106, −4.03229107497633022873496143945, −3.32703175864628220393046312419, −2.73353533009787045767627613771, −2.12856613387019946735417906961, −1.34493680489818486503371968120, −0.43531839274059069948983636995,
0.43531839274059069948983636995, 1.34493680489818486503371968120, 2.12856613387019946735417906961, 2.73353533009787045767627613771, 3.32703175864628220393046312419, 4.03229107497633022873496143945, 4.51050102393655717636124179106, 4.86295202326065104196864761496, 5.42789212116951928797768738883, 5.81185938363869076540861471187, 6.02365212856433686585916068303, 6.72701301081184558305297995727, 7.01612478094549100794464728288, 7.55797087757426684077385479442, 7.76038911057160832928888509258
