| L(s) = 1 | − 2·2-s − 3·3-s + 4-s − 6·5-s + 6·6-s − 4·7-s + 4·9-s + 12·10-s − 6·11-s − 3·12-s − 7·13-s + 8·14-s + 18·15-s + 16-s − 2·17-s − 8·18-s − 2·19-s − 6·20-s + 12·21-s + 12·22-s + 23-s + 18·25-s + 14·26-s − 6·27-s − 4·28-s − 6·29-s − 36·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.73·3-s + 1/2·4-s − 2.68·5-s + 2.44·6-s − 1.51·7-s + 4/3·9-s + 3.79·10-s − 1.80·11-s − 0.866·12-s − 1.94·13-s + 2.13·14-s + 4.64·15-s + 1/4·16-s − 0.485·17-s − 1.88·18-s − 0.458·19-s − 1.34·20-s + 2.61·21-s + 2.55·22-s + 0.208·23-s + 18/5·25-s + 2.74·26-s − 1.15·27-s − 0.755·28-s − 1.11·29-s − 6.57·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66601 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66601 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 66601 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 178 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_4$ | \( 1 + p T + 5 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T - 14 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - T - 21 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 61 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 11 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 68 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - T + 43 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 16 T + 162 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 2 T + 47 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 17 T + 153 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $D_{4}$ | \( 1 + 6 T + 82 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 44 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−15.2699007324, −15.0731854411, −14.6737183190, −13.4531686417, −12.8927202466, −12.7075853280, −12.3435489729, −11.8351450016, −11.3421295562, −11.1808749240, −10.5636344241, −10.1923852207, −9.66120604719, −9.17050548838, −8.68195962370, −7.78861918987, −7.60297684892, −7.44350193266, −6.72509184729, −5.99908818075, −5.37610044386, −4.77033146207, −4.19493706987, −3.41479484402, −2.64568498216, 0, 0, 0,
2.64568498216, 3.41479484402, 4.19493706987, 4.77033146207, 5.37610044386, 5.99908818075, 6.72509184729, 7.44350193266, 7.60297684892, 7.78861918987, 8.68195962370, 9.17050548838, 9.66120604719, 10.1923852207, 10.5636344241, 11.1808749240, 11.3421295562, 11.8351450016, 12.3435489729, 12.7075853280, 12.8927202466, 13.4531686417, 14.6737183190, 15.0731854411, 15.2699007324
