| L(s) = 1 | + 112·13-s + 304·25-s − 1.06e3·37-s − 548·49-s − 1.40e3·61-s − 448·73-s − 2.46e3·97-s − 560·109-s − 1.48e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 948·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 2.38·13-s + 2.43·25-s − 4.72·37-s − 1.59·49-s − 2.93·61-s − 0.718·73-s − 2.57·97-s − 0.492·109-s − 1.11·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.431·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + 0.000326·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.408440091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.408440091\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 152 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 274 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 742 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 28 T + p^{3} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 7376 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 22414 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 29960 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 12542 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 86000 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 155174 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 113566 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 283304 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 34438 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 350 T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 413366 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 283822 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 939038 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 1049494 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 271840 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 616 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.19044393012734946419081014868, −7.11850860167752500922430068525, −6.74569305072667150788826118710, −6.60481943282644243295187715394, −6.27611453710054845904188439965, −6.22923123044801538922027477194, −6.01690580589182594729572899593, −5.45692881413857291213861421213, −5.23258062956057404428112131537, −5.18287639297812592385711966978, −4.98134802660682366656691333574, −4.54964073651829808912779227015, −4.26581859922614565981962294712, −3.83638272607287902164372165744, −3.77817652474254032516056098670, −3.47631522186007107615865370566, −2.97549654534338380511554071239, −2.96431693299985643827471125808, −2.79444657542158545398490952189, −1.96311358558347037278771411274, −1.61036835592199053886844457545, −1.44860806478537460856185995061, −1.34434535288140661233422688678, −0.62966155712764895540975594634, −0.23541224259061507130278464910,
0.23541224259061507130278464910, 0.62966155712764895540975594634, 1.34434535288140661233422688678, 1.44860806478537460856185995061, 1.61036835592199053886844457545, 1.96311358558347037278771411274, 2.79444657542158545398490952189, 2.96431693299985643827471125808, 2.97549654534338380511554071239, 3.47631522186007107615865370566, 3.77817652474254032516056098670, 3.83638272607287902164372165744, 4.26581859922614565981962294712, 4.54964073651829808912779227015, 4.98134802660682366656691333574, 5.18287639297812592385711966978, 5.23258062956057404428112131537, 5.45692881413857291213861421213, 6.01690580589182594729572899593, 6.22923123044801538922027477194, 6.27611453710054845904188439965, 6.60481943282644243295187715394, 6.74569305072667150788826118710, 7.11850860167752500922430068525, 7.19044393012734946419081014868
