| L(s) = 1 | − 4-s + 2·5-s − 2·9-s + 16-s − 2·20-s + 3·25-s + 2·36-s + 4·41-s − 4·45-s − 2·49-s − 2·61-s − 64-s + 2·80-s + 3·81-s − 3·100-s + 4·109-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | − 4-s + 2·5-s − 2·9-s + 16-s − 2·20-s + 3·25-s + 2·36-s + 4·41-s − 4·45-s − 2·49-s − 2·61-s − 64-s + 2·80-s + 3·81-s − 3·100-s + 4·109-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 2·144-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1488400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1488400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.025349882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.025349882\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_1$ | \( ( 1 - T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−9.875654971518710773505388025883, −9.749846535588232864085426772866, −9.125029990230496399821465442003, −9.023786767928471770946365133604, −8.834010062839911382611563390419, −8.172238522086393553631326493402, −7.79157648808659756304414926197, −7.39336182382663985293430643451, −6.51962613114337833237107240642, −6.09061010738157836122811312517, −6.01395249213667920580026302262, −5.62164366532300495213411595071, −5.02931938595896711998591032474, −4.83163237360220268403305872554, −4.16874568524925645536223105850, −3.37758998244806520471841649528, −2.87070139465199709796666070195, −2.55146709303072824258841594428, −1.82577973785851292515864160489, −0.941518076971522480012723485096,
0.941518076971522480012723485096, 1.82577973785851292515864160489, 2.55146709303072824258841594428, 2.87070139465199709796666070195, 3.37758998244806520471841649528, 4.16874568524925645536223105850, 4.83163237360220268403305872554, 5.02931938595896711998591032474, 5.62164366532300495213411595071, 6.01395249213667920580026302262, 6.09061010738157836122811312517, 6.51962613114337833237107240642, 7.39336182382663985293430643451, 7.79157648808659756304414926197, 8.172238522086393553631326493402, 8.834010062839911382611563390419, 9.023786767928471770946365133604, 9.125029990230496399821465442003, 9.749846535588232864085426772866, 9.875654971518710773505388025883
