| L(s) = 1 | − 32·16-s + 52·31-s + 188·61-s − 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | − 2·16-s + 1.67·31-s + 3.08·61-s − 4·121-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 + 0.00578·173-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 + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + 0.00418·239-s + 0.00414·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.666580744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666580744\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 4273 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 56447 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 647 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 503522 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - 6837073 T^{4} + p^{8} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 47 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 31874833 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 16169282 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 7682 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + p^{4} T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_2^3$ | \( 1 - 82132513 T^{4} + p^{8} T^{8} \) |
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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
−8.879284116567907814526998344346, −8.337009814233681948363022096054, −8.142171385826094989718768809993, −8.121907735713472106111504316050, −7.62317130757821839450442928037, −7.32132949113411951999078432820, −7.07007966375889320715472572125, −6.78514664316896286720266661071, −6.54463281342431248781369248283, −6.33456026987217426011316657894, −6.12525509673938296103959745486, −5.58323240504541480017343384588, −5.22791570800650475067459139324, −5.15783637662782680491300358999, −4.71982615641361271262246974519, −4.36338157308747118081352390111, −4.18724580958307321321664442767, −3.69473738151865981957809564848, −3.53181649080284101719209177586, −2.73995763901192192210472662713, −2.57744017340251332822836556338, −2.33687282911949475481194264279, −1.68749466644631709000146762114, −1.09268844859616997300644226410, −0.38727563138447122473997950474,
0.38727563138447122473997950474, 1.09268844859616997300644226410, 1.68749466644631709000146762114, 2.33687282911949475481194264279, 2.57744017340251332822836556338, 2.73995763901192192210472662713, 3.53181649080284101719209177586, 3.69473738151865981957809564848, 4.18724580958307321321664442767, 4.36338157308747118081352390111, 4.71982615641361271262246974519, 5.15783637662782680491300358999, 5.22791570800650475067459139324, 5.58323240504541480017343384588, 6.12525509673938296103959745486, 6.33456026987217426011316657894, 6.54463281342431248781369248283, 6.78514664316896286720266661071, 7.07007966375889320715472572125, 7.32132949113411951999078432820, 7.62317130757821839450442928037, 8.121907735713472106111504316050, 8.142171385826094989718768809993, 8.337009814233681948363022096054, 8.879284116567907814526998344346
