| L(s) = 1 | − 2·5-s + 6·7-s − 2·11-s + 2·13-s + 2·19-s + 2·23-s + 3·25-s + 4·29-s + 6·31-s − 12·35-s + 2·37-s + 4·41-s − 8·43-s + 6·47-s + 13·49-s + 4·55-s − 4·59-s + 10·61-s − 4·65-s + 6·67-s + 4·71-s + 2·73-s − 12·77-s + 8·79-s + 8·83-s + 12·89-s + 12·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 2.26·7-s − 0.603·11-s + 0.554·13-s + 0.458·19-s + 0.417·23-s + 3/5·25-s + 0.742·29-s + 1.07·31-s − 2.02·35-s + 0.328·37-s + 0.624·41-s − 1.21·43-s + 0.875·47-s + 13/7·49-s + 0.539·55-s − 0.520·59-s + 1.28·61-s − 0.496·65-s + 0.733·67-s + 0.474·71-s + 0.234·73-s − 1.36·77-s + 0.900·79-s + 0.878·83-s + 1.27·89-s + 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17139600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.839093866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.839093866\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 24 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 47 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 71 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 107 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 132 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T + 132 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 167 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−8.286186221497427599109269680358, −8.257915665569655269676386070751, −7.88430841568634798803829326183, −7.78566296946842476152463429358, −7.14066174885916695706163759629, −7.00888863311170284146490419710, −6.23226735236534657660091952639, −6.22456825359184981668352813852, −5.37630802114382543692091451618, −5.17549083536370361306094858279, −4.77166249015211022164479138542, −4.68296146508467901807861057523, −4.02708314585323576082983517451, −3.75205009172150650063127011724, −3.18132630340438396823424824375, −2.67342477661446101152202137561, −2.19320259583855989671831305463, −1.66805280637585881445171681880, −1.03658110433159790286683807509, −0.67597643643831452414558733457,
0.67597643643831452414558733457, 1.03658110433159790286683807509, 1.66805280637585881445171681880, 2.19320259583855989671831305463, 2.67342477661446101152202137561, 3.18132630340438396823424824375, 3.75205009172150650063127011724, 4.02708314585323576082983517451, 4.68296146508467901807861057523, 4.77166249015211022164479138542, 5.17549083536370361306094858279, 5.37630802114382543692091451618, 6.22456825359184981668352813852, 6.23226735236534657660091952639, 7.00888863311170284146490419710, 7.14066174885916695706163759629, 7.78566296946842476152463429358, 7.88430841568634798803829326183, 8.257915665569655269676386070751, 8.286186221497427599109269680358
