| L(s) = 1 | − 2·3-s + 3·9-s − 25-s − 4·27-s − 2·49-s + 2·75-s + 5·81-s − 4·83-s + 4·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 2·3-s + 3·9-s − 25-s − 4·27-s − 2·49-s + 2·75-s + 5·81-s − 4·83-s + 4·107-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 4·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4974684765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4974684765\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 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
−8.887484216916158388395060563386, −8.316959596523776567222883685941, −8.162962266682954535427776334129, −7.45970949274410699590909001572, −7.33438178128938753698114340475, −7.03095272785522171683243936835, −6.37512496409794968348021218976, −6.36681335763592780053856351684, −5.75349590667892268663322316178, −5.67554635948981915123587390057, −5.16917982287412399886064201762, −4.70531615339593748017124364836, −4.43371192726024344295222772254, −4.11202120942474389035166806814, −3.47293448985008455850743269577, −3.13723728411488524040092425459, −2.25826098126639690136700412319, −1.74734552763181374110662214763, −1.33672138886155401013222576204, −0.48438635220208810746647840892,
0.48438635220208810746647840892, 1.33672138886155401013222576204, 1.74734552763181374110662214763, 2.25826098126639690136700412319, 3.13723728411488524040092425459, 3.47293448985008455850743269577, 4.11202120942474389035166806814, 4.43371192726024344295222772254, 4.70531615339593748017124364836, 5.16917982287412399886064201762, 5.67554635948981915123587390057, 5.75349590667892268663322316178, 6.36681335763592780053856351684, 6.37512496409794968348021218976, 7.03095272785522171683243936835, 7.33438178128938753698114340475, 7.45970949274410699590909001572, 8.162962266682954535427776334129, 8.316959596523776567222883685941, 8.887484216916158388395060563386
