| L(s) = 1 | − 16·19-s + 88·25-s + 224·43-s − 172·49-s + 112·67-s − 80·73-s − 32·97-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 | − 0.842·19-s + 3.51·25-s + 5.20·43-s − 3.51·49-s + 1.67·67-s − 1.09·73-s − 0.329·97-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(9.791558587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.791558587\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{4} T^{4} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - 416 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 118 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 484 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 470 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1970 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2480 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 362 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 572 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 3406 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 2642 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - 7178 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 7190 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 7946 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 12800 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{4} \) |
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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
−6.25213029851470840540613577004, −6.06021007049268191110787673829, −5.78569694249285007533021258339, −5.50748534319640712521198537438, −5.50390331565036000013617277591, −5.13469269158588508528941950604, −4.82032392940589096702291320367, −4.70247615959883990236831686344, −4.60185322990476760025963789374, −4.33900252176644340000439463241, −4.12414046800898171610769724313, −3.82362464643701532026650416746, −3.72042111279777267969323733992, −3.24398810914439047941812494026, −2.97509715014874881185322111811, −2.86606095734531973228216779945, −2.79415156934707671431108711426, −2.51154941481793096695235016888, −1.93017565991734420888557309383, −1.82087483102853739140150284997, −1.70885930161060218508124616812, −1.08994152759511784126989150370, −0.810542913683926016881312822963, −0.53661356136191090791361848151, −0.51612905327263631671059588161,
0.51612905327263631671059588161, 0.53661356136191090791361848151, 0.810542913683926016881312822963, 1.08994152759511784126989150370, 1.70885930161060218508124616812, 1.82087483102853739140150284997, 1.93017565991734420888557309383, 2.51154941481793096695235016888, 2.79415156934707671431108711426, 2.86606095734531973228216779945, 2.97509715014874881185322111811, 3.24398810914439047941812494026, 3.72042111279777267969323733992, 3.82362464643701532026650416746, 4.12414046800898171610769724313, 4.33900252176644340000439463241, 4.60185322990476760025963789374, 4.70247615959883990236831686344, 4.82032392940589096702291320367, 5.13469269158588508528941950604, 5.50390331565036000013617277591, 5.50748534319640712521198537438, 5.78569694249285007533021258339, 6.06021007049268191110787673829, 6.25213029851470840540613577004
