| L(s) = 1 | − 3·2-s − 4·3-s + 4·4-s − 3·5-s + 12·6-s − 5·7-s − 3·8-s + 7·9-s + 9·10-s − 4·11-s − 16·12-s − 6·13-s + 15·14-s + 12·15-s + 3·16-s − 6·17-s − 21·18-s − 5·19-s − 12·20-s + 20·21-s + 12·22-s + 12·24-s + 25-s + 18·26-s − 4·27-s − 20·28-s − 5·29-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 2.30·3-s + 2·4-s − 1.34·5-s + 4.89·6-s − 1.88·7-s − 1.06·8-s + 7/3·9-s + 2.84·10-s − 1.20·11-s − 4.61·12-s − 1.66·13-s + 4.00·14-s + 3.09·15-s + 3/4·16-s − 1.45·17-s − 4.94·18-s − 1.14·19-s − 2.68·20-s + 4.36·21-s + 2.55·22-s + 2.44·24-s + 1/5·25-s + 3.53·26-s − 0.769·27-s − 3.77·28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62233 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62233 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 62233 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 146 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 5 T + 17 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 40 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 16 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 11 T + 81 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + T + 41 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 15 T + 131 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + p T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 103 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 2 T + 87 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 5 T + 114 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
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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
−15.6032606060, −15.0401891794, −14.7691158996, −13.5004191479, −13.0994143524, −12.6280069341, −12.2112481148, −12.0124360748, −11.3229292907, −10.9651598234, −10.5789156256, −10.2073323014, −9.77767327962, −9.31983445962, −8.69767416992, −8.27309005102, −7.54982726691, −7.12745445612, −6.73390487304, −6.25359064986, −5.44040402640, −5.10272616484, −4.20719486567, −3.36502681375, −2.25072988347, 0, 0, 0,
2.25072988347, 3.36502681375, 4.20719486567, 5.10272616484, 5.44040402640, 6.25359064986, 6.73390487304, 7.12745445612, 7.54982726691, 8.27309005102, 8.69767416992, 9.31983445962, 9.77767327962, 10.2073323014, 10.5789156256, 10.9651598234, 11.3229292907, 12.0124360748, 12.2112481148, 12.6280069341, 13.0994143524, 13.5004191479, 14.7691158996, 15.0401891794, 15.6032606060
