| L(s) = 1 | − 3·2-s − 3·3-s + 4·4-s − 5·5-s + 9·6-s − 6·7-s − 3·8-s + 4·9-s + 15·10-s − 11-s − 12·12-s − 4·13-s + 18·14-s + 15·15-s + 3·16-s − 6·17-s − 12·18-s − 8·19-s − 20·20-s + 18·21-s + 3·22-s − 6·23-s + 9·24-s + 11·25-s + 12·26-s − 6·27-s − 24·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.73·3-s + 2·4-s − 2.23·5-s + 3.67·6-s − 2.26·7-s − 1.06·8-s + 4/3·9-s + 4.74·10-s − 0.301·11-s − 3.46·12-s − 1.10·13-s + 4.81·14-s + 3.87·15-s + 3/4·16-s − 1.45·17-s − 2.82·18-s − 1.83·19-s − 4.47·20-s + 3.92·21-s + 0.639·22-s − 1.25·23-s + 1.83·24-s + 11/5·25-s + 2.35·26-s − 1.15·27-s − 4.53·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61553 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61553 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 61553 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 246 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 6 T + 3 p T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + T - 13 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 2 T - 32 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 + T + 81 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 3 T + 81 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 38 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 82 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + T + 57 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 112 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 40 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + T + 43 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 185 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−15.4433512707, −15.1741237701, −14.8203394574, −13.6318472059, −12.9807482343, −12.7300538848, −12.2465802196, −11.9345413319, −11.4306443861, −10.9941975493, −10.6113343990, −10.1568094247, −9.71550375462, −9.22282764431, −8.75017973488, −8.21320460921, −7.61858596394, −7.26600228894, −6.74168778016, −6.22125539553, −5.76854289515, −4.70883671738, −4.02331398735, −3.57414344466, −2.33412559016, 0, 0, 0,
2.33412559016, 3.57414344466, 4.02331398735, 4.70883671738, 5.76854289515, 6.22125539553, 6.74168778016, 7.26600228894, 7.61858596394, 8.21320460921, 8.75017973488, 9.22282764431, 9.71550375462, 10.1568094247, 10.6113343990, 10.9941975493, 11.4306443861, 11.9345413319, 12.2465802196, 12.7300538848, 12.9807482343, 13.6318472059, 14.8203394574, 15.1741237701, 15.4433512707
