| L(s) = 1 | + 2·7-s − 5·9-s − 11-s − 8·13-s + 12·17-s + 16·19-s − 6·23-s − 25-s − 2·37-s − 3·49-s − 12·53-s − 8·61-s − 10·63-s − 2·67-s + 30·71-s − 8·73-s − 2·77-s + 16·81-s + 12·83-s − 16·91-s + 5·99-s + 36·101-s − 30·113-s + 40·117-s + 24·119-s + 121-s + 127-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 5/3·9-s − 0.301·11-s − 2.21·13-s + 2.91·17-s + 3.67·19-s − 1.25·23-s − 1/5·25-s − 0.328·37-s − 3/7·49-s − 1.64·53-s − 1.02·61-s − 1.25·63-s − 0.244·67-s + 3.56·71-s − 0.936·73-s − 0.227·77-s + 16/9·81-s + 1.31·83-s − 1.67·91-s + 0.502·99-s + 3.58·101-s − 2.82·113-s + 3.69·117-s + 2.20·119-s + 1/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1043504 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1043504 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.770826053\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.770826053\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−8.059521676370594977107455917125, −7.60852684015633966980596452192, −7.58229030462110956552034990365, −6.99893081008274752897764110448, −6.04156470611102555199262281088, −5.82813499505413722087353173382, −5.20861330806238964770088530958, −5.11227132195889762323685903220, −4.87382724180177830554280807243, −3.64578121838069945565500080364, −3.27381985666804460223319703873, −2.98498627498897185681365554589, −2.32017312156527014358680620805, −1.47791555735225284197112666141, −0.63029867339111275019783297438,
0.63029867339111275019783297438, 1.47791555735225284197112666141, 2.32017312156527014358680620805, 2.98498627498897185681365554589, 3.27381985666804460223319703873, 3.64578121838069945565500080364, 4.87382724180177830554280807243, 5.11227132195889762323685903220, 5.20861330806238964770088530958, 5.82813499505413722087353173382, 6.04156470611102555199262281088, 6.99893081008274752897764110448, 7.58229030462110956552034990365, 7.60852684015633966980596452192, 8.059521676370594977107455917125
