| L(s) = 1 | + 28·7-s + 44·11-s + 100·25-s + 490·49-s + 1.23e3·77-s + 146·81-s + 152·107-s + 328·113-s + 1.21e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 2.80e3·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | + 4·7-s + 4·11-s + 4·25-s + 10·49-s + 16·77-s + 1.80·81-s + 1.42·107-s + 2.90·113-s + 10·121-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 + 16·175-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(33.04916187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(33.04916187\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 11 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - 146 T^{4} + p^{8} T^{8} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 13 | $C_2^3$ | \( 1 - 12178 T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 142094 T^{4} + p^{8} T^{8} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 31 | $C_2^3$ | \( 1 + 1378574 T^{4} + p^{8} T^{8} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 + 2255822 T^{4} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 926 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 6675826 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 + 2846 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 13711186 T^{4} + p^{8} T^{8} \) |
| 61 | $C_2^3$ | \( 1 + 12099182 T^{4} + p^{8} T^{8} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 56328014 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 12466 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
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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
−6.90322219326583197961280507722, −6.63076090754721892186070016631, −6.18023196809465826248240270118, −6.16839138729592271747687604273, −5.93348480965602259866691937247, −5.47837946577533208722660749869, −5.36618704009420138766535012429, −5.14068108765951700144920463605, −4.77058466920446620088556807146, −4.57194882778879165985607180782, −4.52433740848850384844306117231, −4.32073031884915755640202991835, −4.30906182610289404629670747156, −3.74008427678792487615628364728, −3.46372039449342272402888365146, −3.20289741964265452223069444323, −3.18939090163165474189233959036, −2.27580770174194441142870712383, −2.23519349330700374696251749283, −2.04405767608905749476009060661, −1.55799519577792651295570905654, −1.42599417821915204199863726690, −0.923437320059324116796012468151, −0.907745116996535980540513052365, −0.887831349199614192755749214926,
0.887831349199614192755749214926, 0.907745116996535980540513052365, 0.923437320059324116796012468151, 1.42599417821915204199863726690, 1.55799519577792651295570905654, 2.04405767608905749476009060661, 2.23519349330700374696251749283, 2.27580770174194441142870712383, 3.18939090163165474189233959036, 3.20289741964265452223069444323, 3.46372039449342272402888365146, 3.74008427678792487615628364728, 4.30906182610289404629670747156, 4.32073031884915755640202991835, 4.52433740848850384844306117231, 4.57194882778879165985607180782, 4.77058466920446620088556807146, 5.14068108765951700144920463605, 5.36618704009420138766535012429, 5.47837946577533208722660749869, 5.93348480965602259866691937247, 6.16839138729592271747687604273, 6.18023196809465826248240270118, 6.63076090754721892186070016631, 6.90322219326583197961280507722
