| L(s) = 1 | − 2·7-s + 4·9-s + 8·23-s + 10·25-s − 8·31-s + 12·41-s + 8·47-s + 3·49-s − 8·63-s − 16·71-s + 16·73-s + 8·79-s + 7·81-s + 16·89-s − 16·97-s − 8·103-s − 20·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 16·161-s + 163-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 4/3·9-s + 1.66·23-s + 2·25-s − 1.43·31-s + 1.87·41-s + 1.16·47-s + 3/7·49-s − 1.00·63-s − 1.89·71-s + 1.87·73-s + 0.900·79-s + 7/9·81-s + 1.69·89-s − 1.62·97-s − 0.788·103-s − 1.88·113-s + 1.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 1.26·161-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12845056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.267948009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.267948009\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 132 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 116 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.082008475537270284404142751852, −8.321280327940950376442793264369, −8.017618252288334884283081878883, −7.44517801858436185174374481451, −7.16489682588057490316213366446, −7.00775169061059145906559740944, −6.63217847665235005625402464253, −6.25063595701885662998339650666, −5.58459689511664056111340658750, −5.50060492935473919520886659098, −4.83696048314413037104726051344, −4.54686375637775899165556274514, −4.19013982021798230106680728237, −3.58370646437304476846122123974, −3.34634259033549967154044644139, −2.70315722275957710489714214388, −2.42902785441597064509547414438, −1.64425235613587955280312590016, −1.07361790115972292758680065743, −0.63832059518674152897906909641,
0.63832059518674152897906909641, 1.07361790115972292758680065743, 1.64425235613587955280312590016, 2.42902785441597064509547414438, 2.70315722275957710489714214388, 3.34634259033549967154044644139, 3.58370646437304476846122123974, 4.19013982021798230106680728237, 4.54686375637775899165556274514, 4.83696048314413037104726051344, 5.50060492935473919520886659098, 5.58459689511664056111340658750, 6.25063595701885662998339650666, 6.63217847665235005625402464253, 7.00775169061059145906559740944, 7.16489682588057490316213366446, 7.44517801858436185174374481451, 8.017618252288334884283081878883, 8.321280327940950376442793264369, 9.082008475537270284404142751852
