| L(s) = 1 | − 2·2-s + 4·3-s + 3·4-s − 8·6-s + 4·7-s − 4·8-s + 7·9-s − 4·11-s + 12·12-s + 2·13-s − 8·14-s + 5·16-s − 14·18-s + 2·19-s + 16·21-s + 8·22-s − 16·24-s + 25-s − 4·26-s + 4·27-s + 12·28-s − 4·31-s − 6·32-s − 16·33-s + 21·36-s − 12·37-s − 4·38-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 2.30·3-s + 3/2·4-s − 3.26·6-s + 1.51·7-s − 1.41·8-s + 7/3·9-s − 1.20·11-s + 3.46·12-s + 0.554·13-s − 2.13·14-s + 5/4·16-s − 3.29·18-s + 0.458·19-s + 3.49·21-s + 1.70·22-s − 3.26·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s + 2.26·28-s − 0.718·31-s − 1.06·32-s − 2.78·33-s + 7/2·36-s − 1.97·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.482590443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.482590443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−15.4802893459, −14.7042188078, −14.5856217610, −14.0366072852, −13.8085645982, −12.8899956802, −12.8337363923, −11.7739202135, −11.3100228916, −10.9484142878, −10.3636633894, −9.67317537122, −9.47750402875, −8.69848461419, −8.33817548631, −8.04714878525, −7.95836653673, −7.12227398289, −6.57312904075, −5.32145208242, −4.92015064778, −3.40344370945, −3.29460314858, −2.08881038737, −1.79579054759,
1.79579054759, 2.08881038737, 3.29460314858, 3.40344370945, 4.92015064778, 5.32145208242, 6.57312904075, 7.12227398289, 7.95836653673, 8.04714878525, 8.33817548631, 8.69848461419, 9.47750402875, 9.67317537122, 10.3636633894, 10.9484142878, 11.3100228916, 11.7739202135, 12.8337363923, 12.8899956802, 13.8085645982, 14.0366072852, 14.5856217610, 14.7042188078, 15.4802893459
