| L(s) = 1 | + 80·19-s − 8·25-s − 160·43-s + 20·49-s + 400·67-s − 80·73-s + 160·97-s + 340·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 460·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 4.21·19-s − 0.319·25-s − 3.72·43-s + 0.408·49-s + 5.97·67-s − 1.09·73-s + 1.64·97-s + 2.80·121-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 − 2.72·169-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 + 0.00448·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.412241922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.412241922\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 170 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 230 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 842 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 332 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 778 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1010 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 1840 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + 982 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 4268 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5810 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 100 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 4682 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 9782 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 2422 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 15680 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 40 T + p^{2} T^{2} )^{4} \) |
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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
−6.24425982317031920733190952157, −5.90115153763228743210634781437, −5.64102413480843861107871154739, −5.63946516508040662483600162842, −5.25442111131909582191539471001, −5.20310637258927422168880035640, −5.04537553364649923159894401258, −4.76092706813331023483617193627, −4.75703264555239349952117794339, −4.21735113280048398557077017268, −3.92003821644616479590952512481, −3.91644348636113494274098393692, −3.40076810202251057683538971862, −3.28921886309282850884478807122, −3.19768900578359794708159765777, −3.15966650804142956670537742611, −2.71983436806357122619580117049, −2.15434033059981774096778030668, −2.06327976465073044908704163989, −2.02139314297924976916718400226, −1.45118393424470884953513040512, −1.07958859717011251383113988173, −0.925977821629592830015472826714, −0.77371257571262982094640833858, −0.17003908418920058019698331097,
0.17003908418920058019698331097, 0.77371257571262982094640833858, 0.925977821629592830015472826714, 1.07958859717011251383113988173, 1.45118393424470884953513040512, 2.02139314297924976916718400226, 2.06327976465073044908704163989, 2.15434033059981774096778030668, 2.71983436806357122619580117049, 3.15966650804142956670537742611, 3.19768900578359794708159765777, 3.28921886309282850884478807122, 3.40076810202251057683538971862, 3.91644348636113494274098393692, 3.92003821644616479590952512481, 4.21735113280048398557077017268, 4.75703264555239349952117794339, 4.76092706813331023483617193627, 5.04537553364649923159894401258, 5.20310637258927422168880035640, 5.25442111131909582191539471001, 5.63946516508040662483600162842, 5.64102413480843861107871154739, 5.90115153763228743210634781437, 6.24425982317031920733190952157
