| L(s) = 1 | + 2-s + 3·3-s + 2·4-s + 3·6-s − 3·7-s + 5·8-s + 6·9-s + 2·11-s + 6·12-s − 2·13-s − 3·14-s + 5·16-s − 8·17-s + 6·18-s − 16·19-s − 9·21-s + 2·22-s + 3·23-s + 15·24-s − 2·26-s + 9·27-s − 6·28-s + 29-s + 10·32-s + 6·33-s − 8·34-s + 12·36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 4-s + 1.22·6-s − 1.13·7-s + 1.76·8-s + 2·9-s + 0.603·11-s + 1.73·12-s − 0.554·13-s − 0.801·14-s + 5/4·16-s − 1.94·17-s + 1.41·18-s − 3.67·19-s − 1.96·21-s + 0.426·22-s + 0.625·23-s + 3.06·24-s − 0.392·26-s + 1.73·27-s − 1.13·28-s + 0.185·29-s + 1.76·32-s + 1.04·33-s − 1.37·34-s + 2·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.719056121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.719056121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 3 T + 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 137 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 6 T - 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−12.79054183804456881792857621252, −12.32700965709325379584105972140, −11.45570073539518445627371394529, −11.04048345028113062837883033782, −10.35775039409588314325214837352, −10.30002848156502976792853398308, −9.499642426053140473710067247109, −8.981709833724890215635177267602, −8.373990530836657260994336089461, −8.338431592470254857090624503554, −7.13187191844188848339622197555, −7.07311713550832796347836079733, −6.51698324827261283119425524520, −6.11466619712105133099479900206, −4.73051919463241812527567815526, −4.24175257649284199512192035834, −4.03678102902236035474044043637, −3.03345464767430081642382528285, −2.15270273663897087433423311516, −2.14335186687034947077085323824,
2.14335186687034947077085323824, 2.15270273663897087433423311516, 3.03345464767430081642382528285, 4.03678102902236035474044043637, 4.24175257649284199512192035834, 4.73051919463241812527567815526, 6.11466619712105133099479900206, 6.51698324827261283119425524520, 7.07311713550832796347836079733, 7.13187191844188848339622197555, 8.338431592470254857090624503554, 8.373990530836657260994336089461, 8.981709833724890215635177267602, 9.499642426053140473710067247109, 10.30002848156502976792853398308, 10.35775039409588314325214837352, 11.04048345028113062837883033782, 11.45570073539518445627371394529, 12.32700965709325379584105972140, 12.79054183804456881792857621252
