L(s) = 1 | + 5·7-s + 10·13-s + 19-s + 5·25-s − 11·31-s − 11·37-s − 26·43-s + 18·49-s − 14·61-s − 5·67-s − 17·73-s − 17·79-s + 50·91-s + 28·97-s + 13·103-s + 19·109-s + 11·121-s + 127-s + 131-s + 5·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 1.88·7-s + 2.77·13-s + 0.229·19-s + 25-s − 1.97·31-s − 1.80·37-s − 3.96·43-s + 18/7·49-s − 1.79·61-s − 0.610·67-s − 1.98·73-s − 1.91·79-s + 5.24·91-s + 2.84·97-s + 1.28·103-s + 1.81·109-s + 121-s + 0.0887·127-s + 0.0873·131-s + 0.433·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.942424221\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.942424221\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
show more | | |
show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.99929450501712945365765238435, −11.76781945541260148277579085538, −11.16649553244544186493239264889, −11.08157249016867279069507421164, −10.37132101945693126350887489407, −10.26770479503104807237374766039, −8.989685743507978045566270894294, −8.891602898263775853725812511253, −8.372007332579292663194765642537, −8.172325327717420668150147378323, −7.13029545854197214861403933751, −7.12169118176407111083321705790, −5.90699610136419423916435487577, −5.89035506414127979385176340765, −4.84025151217890051943393771530, −4.71978140179157600387267003743, −3.46639878577192253208900107072, −3.46622239353313215930627619790, −1.67259348408527152003717601706, −1.54297146000919957214995670622,
1.54297146000919957214995670622, 1.67259348408527152003717601706, 3.46622239353313215930627619790, 3.46639878577192253208900107072, 4.71978140179157600387267003743, 4.84025151217890051943393771530, 5.89035506414127979385176340765, 5.90699610136419423916435487577, 7.12169118176407111083321705790, 7.13029545854197214861403933751, 8.172325327717420668150147378323, 8.372007332579292663194765642537, 8.891602898263775853725812511253, 8.989685743507978045566270894294, 10.26770479503104807237374766039, 10.37132101945693126350887489407, 11.08157249016867279069507421164, 11.16649553244544186493239264889, 11.76781945541260148277579085538, 11.99929450501712945365765238435