| L(s) = 1 | − 3-s − 2·5-s − 7-s + 3·9-s − 2·11-s − 4·13-s + 2·15-s − 3·17-s + 7·19-s + 21-s + 6·23-s + 3·25-s − 8·27-s + 3·29-s − 31-s + 2·33-s + 2·35-s − 13·37-s + 4·39-s − 8·43-s − 6·45-s + 6·47-s − 5·49-s + 3·51-s − 9·53-s + 4·55-s − 7·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 9-s − 0.603·11-s − 1.10·13-s + 0.516·15-s − 0.727·17-s + 1.60·19-s + 0.218·21-s + 1.25·23-s + 3/5·25-s − 1.53·27-s + 0.557·29-s − 0.179·31-s + 0.348·33-s + 0.338·35-s − 2.13·37-s + 0.640·39-s − 1.21·43-s − 0.894·45-s + 0.875·47-s − 5/7·49-s + 0.420·51-s − 1.23·53-s + 0.539·55-s − 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5896012364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5896012364\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 54 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 13 T + 108 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 70 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 9 T + 52 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 5 T + 120 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 70 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 174 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 172 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 14 T + 210 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
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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
−8.568295078862601154141834675733, −8.472033230476338261710956944166, −7.927259861624914424435035609099, −7.40578008556889700755969915125, −7.36768640379511652456414382992, −6.97014231300037966282104508460, −6.64275714487007653546021853807, −6.32481062830683499728239128891, −5.54025206041703473261040644776, −5.22102495264834999842002506697, −5.02250042297340213860114266036, −4.75596054819636685717859538016, −4.05185057878905144686531053649, −3.76857983444227760967653903739, −3.25148845019638194202190276875, −2.90629108538924186160502825272, −2.28642892967872166169380457678, −1.68612129142009338092368317542, −1.03451460064116048696034726887, −0.27529532082538466168583224377,
0.27529532082538466168583224377, 1.03451460064116048696034726887, 1.68612129142009338092368317542, 2.28642892967872166169380457678, 2.90629108538924186160502825272, 3.25148845019638194202190276875, 3.76857983444227760967653903739, 4.05185057878905144686531053649, 4.75596054819636685717859538016, 5.02250042297340213860114266036, 5.22102495264834999842002506697, 5.54025206041703473261040644776, 6.32481062830683499728239128891, 6.64275714487007653546021853807, 6.97014231300037966282104508460, 7.36768640379511652456414382992, 7.40578008556889700755969915125, 7.927259861624914424435035609099, 8.472033230476338261710956944166, 8.568295078862601154141834675733
