| L(s) = 1 | + 16·23-s − 16·25-s + 48·47-s − 8·49-s + 32·71-s − 16·73-s + 8·97-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | + 3.33·23-s − 3.19·25-s + 7.00·47-s − 8/7·49-s + 3.79·71-s − 1.87·73-s + 0.812·97-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.334814344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.334814344\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 148 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
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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
−5.72810406169061293069298865999, −5.69258916045402141201762065949, −5.42050118480344015855886739654, −5.40125864930767125737603720676, −5.20653465278591567684036338619, −4.88873237705608651752637857316, −4.67771885008315949356852498337, −4.40521541998813628132541501514, −4.39288069866896702834278131862, −3.92141511194292064960164999471, −3.78305688164359098616317651395, −3.77834763442584886276964835558, −3.72560343477229900593509817582, −3.05865845718485846931323362306, −2.95186339540425728605374867003, −2.82204764569431329316125914538, −2.65370257639411034436113204450, −2.23051793818484345219107273088, −2.05861198590112527805779050981, −1.90060439038659570335822881819, −1.63200484753431036952306658602, −1.01817779908805260688763133885, −0.868316741748572091798173712139, −0.830122056827369600889606353403, −0.29442274249781570382213466106,
0.29442274249781570382213466106, 0.830122056827369600889606353403, 0.868316741748572091798173712139, 1.01817779908805260688763133885, 1.63200484753431036952306658602, 1.90060439038659570335822881819, 2.05861198590112527805779050981, 2.23051793818484345219107273088, 2.65370257639411034436113204450, 2.82204764569431329316125914538, 2.95186339540425728605374867003, 3.05865845718485846931323362306, 3.72560343477229900593509817582, 3.77834763442584886276964835558, 3.78305688164359098616317651395, 3.92141511194292064960164999471, 4.39288069866896702834278131862, 4.40521541998813628132541501514, 4.67771885008315949356852498337, 4.88873237705608651752637857316, 5.20653465278591567684036338619, 5.40125864930767125737603720676, 5.42050118480344015855886739654, 5.69258916045402141201762065949, 5.72810406169061293069298865999
