| L(s) = 1 | + 3-s + 4·7-s − 2·9-s − 8·13-s + 16·19-s + 4·21-s − 25-s − 5·27-s + 10·31-s − 2·37-s − 8·39-s − 20·43-s − 2·49-s + 16·57-s − 8·61-s − 8·63-s − 2·67-s − 8·73-s − 75-s + 4·79-s + 81-s − 32·91-s + 10·93-s − 14·97-s + 16·103-s + 4·109-s − 2·111-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s − 2.21·13-s + 3.67·19-s + 0.872·21-s − 1/5·25-s − 0.962·27-s + 1.79·31-s − 0.328·37-s − 1.28·39-s − 3.04·43-s − 2/7·49-s + 2.11·57-s − 1.02·61-s − 1.00·63-s − 0.244·67-s − 0.936·73-s − 0.115·75-s + 0.450·79-s + 1/9·81-s − 3.35·91-s + 1.03·93-s − 1.42·97-s + 1.57·103-s + 0.383·109-s − 0.189·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.418807030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418807030\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−11.33128539576622377975036396166, −10.21853205506356381906823281470, −9.813753565180213813561671136388, −9.542494902966876917365028895112, −8.721099137152928932255163369219, −8.132665807481574488057539990847, −7.60852684015633966980596452192, −7.46461921731656872852589989414, −6.54488810595431397607564592743, −5.38389169786521128568224402219, −5.11227132195889762323685903220, −4.66090457644079039135079955202, −3.27381985666804460223319703873, −2.80447621890231958099424449949, −1.61335599959865608201260450974,
1.61335599959865608201260450974, 2.80447621890231958099424449949, 3.27381985666804460223319703873, 4.66090457644079039135079955202, 5.11227132195889762323685903220, 5.38389169786521128568224402219, 6.54488810595431397607564592743, 7.46461921731656872852589989414, 7.60852684015633966980596452192, 8.132665807481574488057539990847, 8.721099137152928932255163369219, 9.542494902966876917365028895112, 9.813753565180213813561671136388, 10.21853205506356381906823281470, 11.33128539576622377975036396166
