| L(s) = 1 | − 8·11-s − 36·17-s − 24·19-s + 46·25-s − 28·41-s − 56·43-s + 34·49-s − 104·59-s + 8·67-s − 132·73-s + 280·83-s − 60·89-s − 28·97-s + 312·107-s − 196·113-s − 194·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 142·169-s + 173-s + ⋯ |
| L(s) = 1 | − 0.727·11-s − 2.11·17-s − 1.26·19-s + 1.83·25-s − 0.682·41-s − 1.30·43-s + 0.693·49-s − 1.76·59-s + 8/67·67-s − 1.80·73-s + 3.37·83-s − 0.674·89-s − 0.288·97-s + 2.91·107-s − 1.73·113-s − 1.60·121-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 + 0.840·169-s + 0.00578·173-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.03359510910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03359510910\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 46 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 34 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 142 T^{2} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 542 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1486 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 898 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 1838 T^{2} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4162 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1262 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 52 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 718 T^{2} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 6946 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 66 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 12226 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 140 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 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
−8.942707034355092813912025595740, −8.697731638327443359063671448394, −8.370181052562010291334944506929, −7.889467957867584528902136002756, −7.50258737066956086574066977931, −6.91496771746120635167912446577, −6.79241488472659275772393882685, −6.22469679367292626025326881674, −6.16535887189092893500864498551, −5.34848685107439339589964499238, −4.95293463466037859879983141279, −4.61554565046261241449438482568, −4.39759605720404394077324318651, −3.65673098220255766062119950968, −3.29073264636413013735897090367, −2.52050534777745948057186738638, −2.41655184693498554699568449575, −1.75178114036350055355328843710, −1.08540858575378585907362876523, −0.04638254913553108880378809455,
0.04638254913553108880378809455, 1.08540858575378585907362876523, 1.75178114036350055355328843710, 2.41655184693498554699568449575, 2.52050534777745948057186738638, 3.29073264636413013735897090367, 3.65673098220255766062119950968, 4.39759605720404394077324318651, 4.61554565046261241449438482568, 4.95293463466037859879983141279, 5.34848685107439339589964499238, 6.16535887189092893500864498551, 6.22469679367292626025326881674, 6.79241488472659275772393882685, 6.91496771746120635167912446577, 7.50258737066956086574066977931, 7.889467957867584528902136002756, 8.370181052562010291334944506929, 8.697731638327443359063671448394, 8.942707034355092813912025595740
