| L(s) = 1 | + 3-s + 9-s + 4·13-s − 8·23-s + 2·25-s + 27-s − 4·37-s + 4·39-s − 8·47-s + 2·49-s + 24·59-s − 4·61-s − 8·69-s − 8·71-s − 12·73-s + 2·75-s + 81-s − 16·83-s − 12·97-s + 8·107-s − 12·109-s − 4·111-s + 4·117-s − 6·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.10·13-s − 1.66·23-s + 2/5·25-s + 0.192·27-s − 0.657·37-s + 0.640·39-s − 1.16·47-s + 2/7·49-s + 3.12·59-s − 0.512·61-s − 0.963·69-s − 0.949·71-s − 1.40·73-s + 0.230·75-s + 1/9·81-s − 1.75·83-s − 1.21·97-s + 0.773·107-s − 1.14·109-s − 0.379·111-s + 0.369·117-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.297359770\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.297359770\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.28955539480983717780048740601, −10.55467990911002305016053439470, −10.06450705240089985497811341595, −9.672774986160731423707906364184, −8.708619271780745558388678055824, −8.596298439519154456295717789464, −7.951693220738384368470547662262, −7.24854452972159581476935000101, −6.61874986053856350598440760272, −5.94967455798711962546753621344, −5.29599759141224461760104533880, −4.25465075696624694903438342673, −3.75305917223549109540389590018, −2.81122173789091023720858389403, −1.66333528881256100507897719111,
1.66333528881256100507897719111, 2.81122173789091023720858389403, 3.75305917223549109540389590018, 4.25465075696624694903438342673, 5.29599759141224461760104533880, 5.94967455798711962546753621344, 6.61874986053856350598440760272, 7.24854452972159581476935000101, 7.951693220738384368470547662262, 8.596298439519154456295717789464, 8.708619271780745558388678055824, 9.672774986160731423707906364184, 10.06450705240089985497811341595, 10.55467990911002305016053439470, 11.28955539480983717780048740601
