| L(s) = 1 | − 2-s + 4-s − 4·5-s − 7-s − 3·8-s + 2·9-s + 4·10-s + 2·11-s + 2·13-s + 14-s + 16-s − 2·18-s + 2·19-s − 4·20-s − 2·22-s + 2·25-s − 2·26-s − 28-s + 3·29-s − 2·31-s + 32-s + 4·35-s + 2·36-s − 3·37-s − 2·38-s + 12·40-s + 8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.78·5-s − 0.377·7-s − 1.06·8-s + 2/3·9-s + 1.26·10-s + 0.603·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.471·18-s + 0.458·19-s − 0.894·20-s − 0.426·22-s + 2/5·25-s − 0.392·26-s − 0.188·28-s + 0.557·29-s − 0.359·31-s + 0.176·32-s + 0.676·35-s + 1/3·36-s − 0.493·37-s − 0.324·38-s + 1.89·40-s + 1.24·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 691 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 691 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2939463892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2939463892\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 691 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 28 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 16 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 3 T + 10 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 10 T + 42 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 5 T + 96 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7 T + 118 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T - 42 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + T + 156 T^{2} + p T^{3} + p^{2} T^{4} \) |
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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
−19.9020086823, −19.4142094782, −19.0743999994, −18.5125889488, −17.8340602584, −17.3776495989, −16.2849893562, −15.9762086755, −15.6646375575, −15.0524826461, −14.3209810526, −13.3902083542, −12.4759502215, −12.0343077422, −11.4304995014, −10.9092022634, −9.82605356051, −9.20536305837, −8.39033934120, −7.68684666296, −6.96753401799, −6.06606216175, −4.33714973357, −3.40174137565,
3.40174137565, 4.33714973357, 6.06606216175, 6.96753401799, 7.68684666296, 8.39033934120, 9.20536305837, 9.82605356051, 10.9092022634, 11.4304995014, 12.0343077422, 12.4759502215, 13.3902083542, 14.3209810526, 15.0524826461, 15.6646375575, 15.9762086755, 16.2849893562, 17.3776495989, 17.8340602584, 18.5125889488, 19.0743999994, 19.4142094782, 19.9020086823
