| L(s) = 1 | + 2-s − 3-s − 2·4-s − 6-s − 5·7-s − 3·8-s − 3·9-s + 11-s + 2·12-s + 2·13-s − 5·14-s + 16-s − 3·18-s − 6·19-s + 5·21-s + 22-s + 7·23-s + 3·24-s − 6·25-s + 2·26-s + 4·27-s + 10·28-s − 4·29-s − 31-s + 2·32-s − 33-s + 6·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 4-s − 0.408·6-s − 1.88·7-s − 1.06·8-s − 9-s + 0.301·11-s + 0.577·12-s + 0.554·13-s − 1.33·14-s + 1/4·16-s − 0.707·18-s − 1.37·19-s + 1.09·21-s + 0.213·22-s + 1.45·23-s + 0.612·24-s − 6/5·25-s + 0.392·26-s + 0.769·27-s + 1.88·28-s − 0.742·29-s − 0.179·31-s + 0.353·32-s − 0.174·33-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10097 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10097 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 6 T + p T^{2} ) \) |
| 439 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 19 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T - 36 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 22 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 50 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 15 T + 124 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 9 T + 104 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 153 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 138 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 3 T - 106 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 130 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−16.9513008850, −16.4221185628, −15.8225950863, −15.2201648520, −14.8978115244, −14.1820455381, −13.7085923928, −13.3210781558, −12.8901393248, −12.5219334795, −11.9993739566, −11.1525316157, −10.9108401409, −10.0283345175, −9.51299850459, −8.91301101904, −8.75574417117, −7.72108401117, −6.78311048492, −6.15971072072, −5.86095935591, −5.12214787157, −4.18376905777, −3.63111467908, −2.81154500052, 0,
2.81154500052, 3.63111467908, 4.18376905777, 5.12214787157, 5.86095935591, 6.15971072072, 6.78311048492, 7.72108401117, 8.75574417117, 8.91301101904, 9.51299850459, 10.0283345175, 10.9108401409, 11.1525316157, 11.9993739566, 12.5219334795, 12.8901393248, 13.3210781558, 13.7085923928, 14.1820455381, 14.8978115244, 15.2201648520, 15.8225950863, 16.4221185628, 16.9513008850
