L(s) = 1 | + 2·4-s − 7-s + 4·13-s + 7·19-s + 5·25-s − 2·28-s − 11·31-s + 10·37-s − 26·43-s − 6·49-s + 8·52-s + 13·61-s − 8·64-s + 16·67-s + 7·73-s + 14·76-s + 4·79-s − 4·91-s + 10·97-s + 10·100-s − 20·103-s + 19·109-s + 11·121-s − 22·124-s + 127-s + 131-s − 7·133-s + ⋯ |
L(s) = 1 | + 4-s − 0.377·7-s + 1.10·13-s + 1.60·19-s + 25-s − 0.377·28-s − 1.97·31-s + 1.64·37-s − 3.96·43-s − 6/7·49-s + 1.10·52-s + 1.66·61-s − 64-s + 1.95·67-s + 0.819·73-s + 1.60·76-s + 0.450·79-s − 0.419·91-s + 1.01·97-s + 100-s − 1.97·103-s + 1.81·109-s + 121-s − 1.97·124-s + 0.0887·127-s + 0.0873·131-s − 0.606·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35721 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.631534109\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.631534109\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−12.80444859314965608835283090665, −12.34642365760301109895533903481, −11.56133513843806040725467735973, −11.28553459171877990901760917024, −11.19686013625458801930042760087, −10.38411073186735632279211533145, −9.795318746449690797632937535154, −9.513890445895080945335261376144, −8.710205457721235052209137861185, −8.322709809806686420757083655108, −7.60468031972361752969737633225, −7.09273544245850110200423430314, −6.56156358666153370563150050580, −6.21676046051896484871812868536, −5.31191683311091176991980917918, −4.95364039631166980593041056903, −3.62604658009176267743390914269, −3.40216835495597057193017243085, −2.39735138305506796527866923784, −1.37407463838619250781379380582,
1.37407463838619250781379380582, 2.39735138305506796527866923784, 3.40216835495597057193017243085, 3.62604658009176267743390914269, 4.95364039631166980593041056903, 5.31191683311091176991980917918, 6.21676046051896484871812868536, 6.56156358666153370563150050580, 7.09273544245850110200423430314, 7.60468031972361752969737633225, 8.322709809806686420757083655108, 8.710205457721235052209137861185, 9.513890445895080945335261376144, 9.795318746449690797632937535154, 10.38411073186735632279211533145, 11.19686013625458801930042760087, 11.28553459171877990901760917024, 11.56133513843806040725467735973, 12.34642365760301109895533903481, 12.80444859314965608835283090665