| L(s) = 1 | − 5·7-s − 10·25-s + 2·37-s − 16·43-s + 18·49-s + 22·67-s − 26·79-s − 4·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s + 50·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 2·25-s + 0.328·37-s − 2.43·43-s + 18/7·49-s + 2.68·67-s − 2.92·79-s − 0.383·109-s + 2·121-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 + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + 0.0760·173-s + 3.77·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8108374221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8108374221\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−10.32175231359791411584049196486, −9.893086223222024123647772416995, −9.889813406078335955541434941801, −9.453109426050750789327109798882, −8.913751522216197291076461934527, −8.382134965287589616455486077094, −8.110573864603345765777075064195, −7.40031489141267915017225163522, −7.03622273857576276276741062693, −6.61656830984141797103630660508, −6.20130449939886660425278286750, −5.71078796891535215951667332307, −5.39362291706783301781115890897, −4.56272641846991815272137540972, −4.03543179677870772489110320726, −3.48618687265904621824860114667, −3.16107985935435432608510306972, −2.41460040215232219778713294450, −1.72953084860175627500825651548, −0.44567927555889145076557034814,
0.44567927555889145076557034814, 1.72953084860175627500825651548, 2.41460040215232219778713294450, 3.16107985935435432608510306972, 3.48618687265904621824860114667, 4.03543179677870772489110320726, 4.56272641846991815272137540972, 5.39362291706783301781115890897, 5.71078796891535215951667332307, 6.20130449939886660425278286750, 6.61656830984141797103630660508, 7.03622273857576276276741062693, 7.40031489141267915017225163522, 8.110573864603345765777075064195, 8.382134965287589616455486077094, 8.913751522216197291076461934527, 9.453109426050750789327109798882, 9.889813406078335955541434941801, 9.893086223222024123647772416995, 10.32175231359791411584049196486
