| L(s) = 1 | + 20·13-s + 48·25-s − 48·37-s − 98·49-s + 240·61-s + 192·73-s − 288·97-s + 364·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.53·13-s + 1.91·25-s − 1.29·37-s − 2·49-s + 3.93·61-s + 2.63·73-s − 2.96·97-s + 3.33·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 0.224·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.919844595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.919844595\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 48 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 480 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 1680 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 1440 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 5040 T^{2} + p^{4} T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 120 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 96 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12480 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 144 T + p^{2} T^{2} )^{2} \) |
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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
−8.829474484440899517734602056623, −8.568708579628585631455101596033, −8.291171142127193147711532692932, −8.160307726540877938062077772303, −7.34560988493813313265906340013, −7.04760156382425246952166526087, −6.61893045178887135210063251452, −6.48358926373150078964595586580, −5.87907337304749731098985815398, −5.45792172537949044686615948224, −5.00608862267483122945426146925, −4.79772981479826657594582065140, −3.98934704598245933147996809130, −3.78840346633234892473676444728, −3.23117559141155859592590978611, −2.91738824568286215836848926407, −2.12276037148964981375399013133, −1.70719786381970485182183148789, −0.966529855225046453087715220503, −0.58609097670388090670597513836,
0.58609097670388090670597513836, 0.966529855225046453087715220503, 1.70719786381970485182183148789, 2.12276037148964981375399013133, 2.91738824568286215836848926407, 3.23117559141155859592590978611, 3.78840346633234892473676444728, 3.98934704598245933147996809130, 4.79772981479826657594582065140, 5.00608862267483122945426146925, 5.45792172537949044686615948224, 5.87907337304749731098985815398, 6.48358926373150078964595586580, 6.61893045178887135210063251452, 7.04760156382425246952166526087, 7.34560988493813313265906340013, 8.160307726540877938062077772303, 8.291171142127193147711532692932, 8.568708579628585631455101596033, 8.829474484440899517734602056623
