| L(s) = 1 | − 2·2-s − 3-s + 3·4-s + 2·5-s + 2·6-s − 6·7-s − 4·8-s − 9-s − 4·10-s − 2·11-s − 3·12-s + 3·13-s + 12·14-s − 2·15-s + 5·16-s + 7·17-s + 2·18-s − 2·19-s + 6·20-s + 6·21-s + 4·22-s − 10·23-s + 4·24-s + 3·25-s − 6·26-s − 18·28-s + 4·29-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s − 2.26·7-s − 1.41·8-s − 1/3·9-s − 1.26·10-s − 0.603·11-s − 0.866·12-s + 0.832·13-s + 3.20·14-s − 0.516·15-s + 5/4·16-s + 1.69·17-s + 0.471·18-s − 0.458·19-s + 1.34·20-s + 1.30·21-s + 0.852·22-s − 2.08·23-s + 0.816·24-s + 3/5·25-s − 1.17·26-s − 3.40·28-s + 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22372900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22372900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 43 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 24 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 42 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 10 T + 54 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 49 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 84 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 3 T + 104 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 126 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T + 142 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 21 T + 248 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 3 T + 144 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 3 T + 156 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 10 T + 38 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11 T + 170 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 18 T + 258 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.007580079804110809689623253868, −7.993898618017887215604906095238, −7.47980034774025462153338626961, −6.86369039113355235338629589299, −6.50926651732781921033079825950, −6.50607968174673415436461760303, −5.98406858106508623749872215234, −5.86453928052456982023723479895, −5.39471985859703678539391523503, −5.12363305254701896738383619826, −4.21426822196787694833387758925, −3.80536764395053676916210707990, −3.30863542701167587921739800369, −3.04580339390244720196042263481, −2.56023467130000419456649096353, −2.15012166554226177636428758754, −1.42174393012231609968400719460, −0.986694571237062052272526580458, 0, 0,
0.986694571237062052272526580458, 1.42174393012231609968400719460, 2.15012166554226177636428758754, 2.56023467130000419456649096353, 3.04580339390244720196042263481, 3.30863542701167587921739800369, 3.80536764395053676916210707990, 4.21426822196787694833387758925, 5.12363305254701896738383619826, 5.39471985859703678539391523503, 5.86453928052456982023723479895, 5.98406858106508623749872215234, 6.50607968174673415436461760303, 6.50926651732781921033079825950, 6.86369039113355235338629589299, 7.47980034774025462153338626961, 7.993898618017887215604906095238, 8.007580079804110809689623253868
