| L(s) = 1 | − 2·3-s − 2·5-s + 3·9-s + 2·11-s + 2·13-s + 4·15-s + 4·17-s + 4·19-s + 8·23-s − 4·25-s − 4·27-s + 8·31-s − 4·33-s + 8·37-s − 4·39-s − 6·41-s − 8·43-s − 6·45-s + 6·47-s − 8·51-s + 20·53-s − 4·55-s − 8·57-s − 26·59-s − 4·61-s − 4·65-s + 20·67-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 9-s + 0.603·11-s + 0.554·13-s + 1.03·15-s + 0.970·17-s + 0.917·19-s + 1.66·23-s − 4/5·25-s − 0.769·27-s + 1.43·31-s − 0.696·33-s + 1.31·37-s − 0.640·39-s − 0.937·41-s − 1.21·43-s − 0.894·45-s + 0.875·47-s − 1.12·51-s + 2.74·53-s − 0.539·55-s − 1.05·57-s − 3.38·59-s − 0.512·61-s − 0.496·65-s + 2.44·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58430736 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.285113117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.285113117\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T - 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 26 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 78 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 88 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 26 T + 284 T^{2} + 26 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 222 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 14 T + 116 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 20 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6 T + 160 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.904696133387382920601185164506, −7.77152843674603387036747726129, −7.16700630810904931642192852650, −7.10452097416777523923980011322, −6.58396165623470241685486679285, −6.41777312565683030939578173483, −5.82412222586304801209055193485, −5.74062457578461743406329773023, −5.12884698935276346411162174747, −5.03078856656000555328252644077, −4.51679823689375048624717539146, −4.16100237133683056497631379124, −3.72352403892414804948529716609, −3.51310115297367884254414380209, −2.95170243566927728413181865760, −2.63205757969519888375762801463, −1.78179794534730876657303852285, −1.35006073233233589063754025141, −0.802092637331039509163911519313, −0.57852670315661870629138675603,
0.57852670315661870629138675603, 0.802092637331039509163911519313, 1.35006073233233589063754025141, 1.78179794534730876657303852285, 2.63205757969519888375762801463, 2.95170243566927728413181865760, 3.51310115297367884254414380209, 3.72352403892414804948529716609, 4.16100237133683056497631379124, 4.51679823689375048624717539146, 5.03078856656000555328252644077, 5.12884698935276346411162174747, 5.74062457578461743406329773023, 5.82412222586304801209055193485, 6.41777312565683030939578173483, 6.58396165623470241685486679285, 7.10452097416777523923980011322, 7.16700630810904931642192852650, 7.77152843674603387036747726129, 7.904696133387382920601185164506
