| L(s) = 1 | − 2·2-s − 2·3-s − 4-s + 4·6-s + 8·8-s + 3·9-s + 2·11-s + 2·12-s − 9·13-s − 7·16-s − 17-s − 6·18-s − 2·19-s − 4·22-s − 16·24-s + 18·26-s − 4·27-s + 11·29-s + 10·31-s − 14·32-s − 4·33-s + 2·34-s − 3·36-s − 5·37-s + 4·38-s + 18·39-s − 5·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s − 1/2·4-s + 1.63·6-s + 2.82·8-s + 9-s + 0.603·11-s + 0.577·12-s − 2.49·13-s − 7/4·16-s − 0.242·17-s − 1.41·18-s − 0.458·19-s − 0.852·22-s − 3.26·24-s + 3.53·26-s − 0.769·27-s + 2.04·29-s + 1.79·31-s − 2.47·32-s − 0.696·33-s + 0.342·34-s − 1/2·36-s − 0.821·37-s + 0.648·38-s + 2.88·39-s − 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3515625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3515625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 9 T + 45 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 23 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 3 p T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 10 T + 82 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 5 T + 49 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 5 T + 87 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 111 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - T + 121 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 146 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 5 T + 121 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 13 T + 219 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11 T + 213 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−9.139631341288759573391165811585, −8.730264778061087714549711535863, −8.189373589478111696146722531933, −8.048192055739203630957445198756, −7.57759296941742729926394613199, −7.08906283646487937755354944775, −6.67003064948681231442910331723, −6.62479377385035413489690622074, −5.72969960009539445483473881836, −5.28833856937473056467156442622, −4.82239830035640573639637618657, −4.71070563011167120070767177093, −4.24858665355340449188517398080, −3.87964560450793591322330078854, −2.80688919068116824546794244027, −2.41606283400103252543369505909, −1.39858431609530956214490230704, −1.13868330344096009920645055642, 0, 0,
1.13868330344096009920645055642, 1.39858431609530956214490230704, 2.41606283400103252543369505909, 2.80688919068116824546794244027, 3.87964560450793591322330078854, 4.24858665355340449188517398080, 4.71070563011167120070767177093, 4.82239830035640573639637618657, 5.28833856937473056467156442622, 5.72969960009539445483473881836, 6.62479377385035413489690622074, 6.67003064948681231442910331723, 7.08906283646487937755354944775, 7.57759296941742729926394613199, 8.048192055739203630957445198756, 8.189373589478111696146722531933, 8.730264778061087714549711535863, 9.139631341288759573391165811585
