| L(s) = 1 | + 2-s + 6·5-s − 7-s − 8-s + 6·10-s + 6·11-s − 5·13-s − 14-s − 16-s + 3·17-s − 5·19-s + 6·22-s + 6·23-s + 17·25-s − 5·26-s − 3·29-s + 4·31-s + 3·34-s − 6·35-s + 7·37-s − 5·38-s − 6·40-s − 9·41-s − 11·43-s + 6·46-s − 6·49-s + 17·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 2.68·5-s − 0.377·7-s − 0.353·8-s + 1.89·10-s + 1.80·11-s − 1.38·13-s − 0.267·14-s − 1/4·16-s + 0.727·17-s − 1.14·19-s + 1.27·22-s + 1.25·23-s + 17/5·25-s − 0.980·26-s − 0.557·29-s + 0.718·31-s + 0.514·34-s − 1.01·35-s + 1.15·37-s − 0.811·38-s − 0.948·40-s − 1.40·41-s − 1.67·43-s + 0.884·46-s − 6/7·49-s + 2.40·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.588272311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.588272311\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5 T + 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 11 T + 48 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 3 T - 74 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 136 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - T - 96 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−11.79266121519492507808887655519, −11.24405396601319947006869578317, −10.39886497102759487159366782188, −10.11373510764729176653577055372, −9.647416055390662117375724183757, −9.619018142985817774990648318467, −8.818895695303352818051336305808, −8.812875209103618643700518907390, −7.80083606031369702444328819105, −7.01373435114625526197951900274, −6.47269556162318140505526488840, −6.41502720249413499461459530719, −5.86709639487708148459070887951, −5.10149594255908444749909301453, −5.04012728088693649310951558838, −4.18234332028656299378009613305, −3.38853552349609335559005879725, −2.69409037696983255698842367186, −2.01824613382574565182518686881, −1.38123911080229985451394679110,
1.38123911080229985451394679110, 2.01824613382574565182518686881, 2.69409037696983255698842367186, 3.38853552349609335559005879725, 4.18234332028656299378009613305, 5.04012728088693649310951558838, 5.10149594255908444749909301453, 5.86709639487708148459070887951, 6.41502720249413499461459530719, 6.47269556162318140505526488840, 7.01373435114625526197951900274, 7.80083606031369702444328819105, 8.812875209103618643700518907390, 8.818895695303352818051336305808, 9.619018142985817774990648318467, 9.647416055390662117375724183757, 10.11373510764729176653577055372, 10.39886497102759487159366782188, 11.24405396601319947006869578317, 11.79266121519492507808887655519
