L(s) = 1 | + 4·7-s + 2·13-s − 4·19-s − 7·25-s + 16·31-s + 14·37-s − 4·43-s − 2·49-s + 14·61-s + 20·67-s − 14·73-s + 4·79-s + 8·91-s + 4·97-s + 16·103-s − 22·109-s − 10·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | + 1.51·7-s + 0.554·13-s − 0.917·19-s − 7/5·25-s + 2.87·31-s + 2.30·37-s − 0.609·43-s − 2/7·49-s + 1.79·61-s + 2.44·67-s − 1.63·73-s + 0.450·79-s + 0.838·91-s + 0.406·97-s + 1.57·103-s − 2.10·109-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.027660741\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.027660741\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 151 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.210131723990394256438834272000, −8.200737600393406002461518862507, −7.72044542760917893777194766109, −7.58842793464416358609597272811, −6.79927285251474469747118325501, −6.62387078398074566460064266724, −6.10780284996259074793155577325, −6.09368272828119035452445046318, −5.34124997281434532057141592253, −5.15961748232879630934924578638, −4.57883917701246067552116005005, −4.46471926198148358606993603713, −3.92026424164421491252919970408, −3.75105597869863896297198484355, −2.79108403819916184967378229983, −2.73265518837680253266345736382, −2.01833638152351395471032114650, −1.73298483451668299840258630058, −1.05490170206817799520388610206, −0.60555317707886083059899131770,
0.60555317707886083059899131770, 1.05490170206817799520388610206, 1.73298483451668299840258630058, 2.01833638152351395471032114650, 2.73265518837680253266345736382, 2.79108403819916184967378229983, 3.75105597869863896297198484355, 3.92026424164421491252919970408, 4.46471926198148358606993603713, 4.57883917701246067552116005005, 5.15961748232879630934924578638, 5.34124997281434532057141592253, 6.09368272828119035452445046318, 6.10780284996259074793155577325, 6.62387078398074566460064266724, 6.79927285251474469747118325501, 7.58842793464416358609597272811, 7.72044542760917893777194766109, 8.200737600393406002461518862507, 8.210131723990394256438834272000