| L(s) = 1 | − 6·5-s + 4·7-s − 5·9-s − 11-s + 16·19-s + 17·25-s − 24·35-s − 2·37-s − 20·43-s + 30·45-s − 2·49-s − 12·53-s + 6·55-s − 20·63-s − 4·77-s + 4·79-s + 16·81-s + 12·83-s − 18·89-s − 96·95-s − 14·97-s + 5·99-s + 12·107-s − 30·113-s + 121-s − 18·125-s + 127-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 1.51·7-s − 5/3·9-s − 0.301·11-s + 3.67·19-s + 17/5·25-s − 4.05·35-s − 0.328·37-s − 3.04·43-s + 4.47·45-s − 2/7·49-s − 1.64·53-s + 0.809·55-s − 2.51·63-s − 0.455·77-s + 0.450·79-s + 16/9·81-s + 1.31·83-s − 1.90·89-s − 9.84·95-s − 1.42·97-s + 0.502·99-s + 1.16·107-s − 2.82·113-s + 1/11·121-s − 1.60·125-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85184 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85184 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 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
−9.422536869125018616649806813738, −8.605213066943072523475490546148, −8.132665807481574488057539990847, −8.106469985083330538380333325542, −7.60852684015633966980596452192, −7.26152152624268428704712414891, −6.49340012695586359351910248567, −5.43339609963471466451709551596, −5.11227132195889762323685903220, −4.79469791447770690232597181563, −3.82876273665516882094535074726, −3.27381985666804460223319703873, −3.00443596972753636527090013772, −1.36947441810401211657481497357, 0,
1.36947441810401211657481497357, 3.00443596972753636527090013772, 3.27381985666804460223319703873, 3.82876273665516882094535074726, 4.79469791447770690232597181563, 5.11227132195889762323685903220, 5.43339609963471466451709551596, 6.49340012695586359351910248567, 7.26152152624268428704712414891, 7.60852684015633966980596452192, 8.106469985083330538380333325542, 8.132665807481574488057539990847, 8.605213066943072523475490546148, 9.422536869125018616649806813738
