| L(s) = 1 | − 3·2-s − 2·3-s + 4·4-s + 6·6-s − 3·7-s − 3·8-s − 4·11-s − 8·12-s + 13-s + 9·14-s + 3·16-s − 5·17-s + 3·19-s + 6·21-s + 12·22-s + 7·23-s + 6·24-s + 7·25-s − 3·26-s + 2·27-s − 12·28-s − 4·29-s + 12·31-s − 6·32-s + 8·33-s + 15·34-s + 12·37-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 2·4-s + 2.44·6-s − 1.13·7-s − 1.06·8-s − 1.20·11-s − 2.30·12-s + 0.277·13-s + 2.40·14-s + 3/4·16-s − 1.21·17-s + 0.688·19-s + 1.30·21-s + 2.55·22-s + 1.45·23-s + 1.22·24-s + 7/5·25-s − 0.588·26-s + 0.384·27-s − 2.26·28-s − 0.742·29-s + 2.15·31-s − 1.06·32-s + 1.39·33-s + 2.57·34-s + 1.97·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36389 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36389 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 36389 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 130 T + p T^{2} ) \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 4 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 21 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T - 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 44 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 39 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 5 T + 31 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 5 T + 114 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 2 T - 31 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 15 T + 131 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 97 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 2 T - 8 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 61 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 60 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 217 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
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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
−15.5697166613, −15.0932241686, −14.7541310954, −13.6221250684, −13.3378473916, −13.0160660208, −12.3661222963, −11.7964412304, −11.3198443842, −10.8981838735, −10.4231839620, −10.1453317784, −9.52944457977, −9.07875880060, −8.72199306754, −8.15556348375, −7.54081079928, −7.03120954229, −6.23511170417, −6.13887048316, −5.10557084190, −4.69688683573, −3.22101794447, −2.67735565214, −0.984544750732, 0,
0.984544750732, 2.67735565214, 3.22101794447, 4.69688683573, 5.10557084190, 6.13887048316, 6.23511170417, 7.03120954229, 7.54081079928, 8.15556348375, 8.72199306754, 9.07875880060, 9.52944457977, 10.1453317784, 10.4231839620, 10.8981838735, 11.3198443842, 11.7964412304, 12.3661222963, 13.0160660208, 13.3378473916, 13.6221250684, 14.7541310954, 15.0932241686, 15.5697166613
