| L(s) = 1 | − 6·4-s + 16·7-s − 10·9-s + 20·16-s − 36·25-s − 96·28-s − 16·31-s + 60·36-s − 36·49-s − 160·63-s − 24·64-s − 24·73-s + 496·79-s + 19·81-s + 472·97-s + 216·100-s − 432·103-s + 320·112-s − 340·121-s + 96·124-s + 127-s + 131-s + 137-s + 139-s − 200·144-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 16/7·7-s − 1.11·9-s + 5/4·16-s − 1.43·25-s − 3.42·28-s − 0.516·31-s + 5/3·36-s − 0.734·49-s − 2.53·63-s − 3/8·64-s − 0.328·73-s + 6.27·79-s + 0.234·81-s + 4.86·97-s + 2.15·100-s − 4.19·103-s + 20/7·112-s − 2.80·121-s + 0.774·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 1.38·144-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331776 ^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6007648398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6007648398\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 + 3 p T^{2} + p^{4} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 10 T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 226 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 354 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 694 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 162 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1394 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2466 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 3670 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 + 3026 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4650 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 1630 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 6710 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2}( 1 + 110 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 124 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 13770 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 4866 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 118 T + p^{2} T^{2} )^{4} \) |
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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
−13.30585491727059322223027458989, −13.06787330122284560108302223006, −12.65300169127315327592621779684, −11.94041167474267908976916677818, −11.89746978123167618675420931678, −11.67471531999220073326774440554, −11.04438544989209094837587220671, −10.79036843865912096372979466727, −10.72062752973857449709958500144, −9.846670408141429861209173533844, −9.544275704799949303938687941939, −9.292175672440491200167963865656, −8.798698392185051621009315897507, −8.351408340978118145097745896039, −8.122046395595495303490504147548, −7.82115244114720862878575879135, −7.55293857840674426208737803386, −6.52670727400072296712392964723, −6.14447145950804349033210209199, −5.38400542037203543276090817067, −4.98220427408142433639761568124, −4.89793294545700987547852631759, −4.05866501531734767507822894176, −3.45864190635624545873021523023, −1.99519762672624888677905786603,
1.99519762672624888677905786603, 3.45864190635624545873021523023, 4.05866501531734767507822894176, 4.89793294545700987547852631759, 4.98220427408142433639761568124, 5.38400542037203543276090817067, 6.14447145950804349033210209199, 6.52670727400072296712392964723, 7.55293857840674426208737803386, 7.82115244114720862878575879135, 8.122046395595495303490504147548, 8.351408340978118145097745896039, 8.798698392185051621009315897507, 9.292175672440491200167963865656, 9.544275704799949303938687941939, 9.846670408141429861209173533844, 10.72062752973857449709958500144, 10.79036843865912096372979466727, 11.04438544989209094837587220671, 11.67471531999220073326774440554, 11.89746978123167618675420931678, 11.94041167474267908976916677818, 12.65300169127315327592621779684, 13.06787330122284560108302223006, 13.30585491727059322223027458989
