| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 2·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s − 4·10-s − 4·11-s − 6·12-s − 4·14-s + 4·15-s + 5·16-s + 4·17-s + 6·18-s + 2·19-s − 6·20-s + 4·21-s − 8·22-s − 8·24-s − 4·25-s − 4·27-s − 6·28-s − 6·29-s + 8·30-s + 2·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 0.894·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.26·10-s − 1.20·11-s − 1.73·12-s − 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.970·17-s + 1.41·18-s + 0.458·19-s − 1.34·20-s + 0.872·21-s − 1.70·22-s − 1.63·24-s − 4/5·25-s − 0.769·27-s − 1.13·28-s − 1.11·29-s + 1.46·30-s + 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50381604 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 2 T - 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 80 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 84 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 83 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 14 T + 143 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 20 T + 219 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 124 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 234 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T - 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 175 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 192 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.60491188952005286021430898257, −7.58145509631131023494676008189, −6.78944660457468836743577281520, −6.73199870497150311580813820600, −6.08929084243319816379316147130, −6.01550537557317169249446694808, −5.53776996053441100246173439024, −5.39864750948483400149937065416, −4.99880740223038422986682391697, −4.49083805521704451209780337618, −4.16418489296889987459682572543, −3.98807138854505203231244587961, −3.33888421009879985532914137220, −3.21240106183492532178608480583, −2.52133487055481954470454049275, −2.41179960947626628687517788851, −1.38733439384469632512183222708, −1.17294995297692128885344185238, 0, 0,
1.17294995297692128885344185238, 1.38733439384469632512183222708, 2.41179960947626628687517788851, 2.52133487055481954470454049275, 3.21240106183492532178608480583, 3.33888421009879985532914137220, 3.98807138854505203231244587961, 4.16418489296889987459682572543, 4.49083805521704451209780337618, 4.99880740223038422986682391697, 5.39864750948483400149937065416, 5.53776996053441100246173439024, 6.01550537557317169249446694808, 6.08929084243319816379316147130, 6.73199870497150311580813820600, 6.78944660457468836743577281520, 7.58145509631131023494676008189, 7.60491188952005286021430898257
