| L(s) = 1 | + 24·17-s + 4·25-s − 264·41-s + 68·49-s + 232·73-s + 408·89-s + 104·97-s − 264·113-s − 460·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 580·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 1.41·17-s + 4/25·25-s − 6.43·41-s + 1.38·49-s + 3.17·73-s + 4.58·89-s + 1.07·97-s − 2.33·113-s − 3.80·121-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 + 3.43·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.235192019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.235192019\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 230 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 290 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 482 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 1250 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 898 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2690 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 66 T + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + 2726 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2114 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2494 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 5990 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 670 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2630 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 4318 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 58 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 12226 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 10310 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 102 T + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{4} \) |
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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
−7.65347401402976367445488380801, −7.19613283116764976996914135715, −7.14485537768940802741614162243, −6.59053437397292248751262198451, −6.46448751373297539329963678388, −6.43444611613718735024981344997, −6.41989645775456242828942133132, −5.61032980123761353348683253873, −5.43945339064436652221843965253, −5.26952995299851617965903852198, −5.13115607068389196842923595779, −5.00492144507104612943197577151, −4.52094969889126357453689375098, −4.24101081885450694769848992197, −3.76645510790283891557342393443, −3.60138317685090851828448381269, −3.46612964115257328286566156286, −3.08504669699698398390656221693, −2.91587225445365071247724793670, −2.16355512716629232291942038257, −2.15624649162872048585836105981, −1.61416481305473965127740429835, −1.38698840451178092756608876823, −0.70458366384506180523000426863, −0.39642264749604235473629967220,
0.39642264749604235473629967220, 0.70458366384506180523000426863, 1.38698840451178092756608876823, 1.61416481305473965127740429835, 2.15624649162872048585836105981, 2.16355512716629232291942038257, 2.91587225445365071247724793670, 3.08504669699698398390656221693, 3.46612964115257328286566156286, 3.60138317685090851828448381269, 3.76645510790283891557342393443, 4.24101081885450694769848992197, 4.52094969889126357453689375098, 5.00492144507104612943197577151, 5.13115607068389196842923595779, 5.26952995299851617965903852198, 5.43945339064436652221843965253, 5.61032980123761353348683253873, 6.41989645775456242828942133132, 6.43444611613718735024981344997, 6.46448751373297539329963678388, 6.59053437397292248751262198451, 7.14485537768940802741614162243, 7.19613283116764976996914135715, 7.65347401402976367445488380801
