| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 3·7-s + 8-s − 2·9-s − 10-s + 4·11-s − 12-s − 3·13-s − 3·14-s + 15-s + 16-s − 17-s − 2·18-s + 5·19-s − 20-s + 3·21-s + 4·22-s + 8·23-s − 24-s + 25-s − 3·26-s + 5·27-s − 3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.832·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.471·18-s + 1.14·19-s − 0.223·20-s + 0.654·21-s + 0.852·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.588·26-s + 0.962·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.075898030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.075898030\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.87399930440417, −14.07582899575711, −13.87354483112647, −13.20765239732595, −12.48693240162143, −12.19444746511934, −11.82352635463387, −11.25822540835418, −10.75576259882908, −10.15497348593496, −9.367767463524065, −9.131700357396184, −8.430877422089723, −7.458298255694791, −7.123558185466566, −6.549478659911385, −6.097822966633072, −5.450034404657987, −4.861807126720060, −4.325527865758614, −3.420386762152525, −3.137185123775362, −2.473745815296430, −1.271652845578786, −0.5206171715846677,
0.5206171715846677, 1.271652845578786, 2.473745815296430, 3.137185123775362, 3.420386762152525, 4.325527865758614, 4.861807126720060, 5.450034404657987, 6.097822966633072, 6.549478659911385, 7.123558185466566, 7.458298255694791, 8.430877422089723, 9.131700357396184, 9.367767463524065, 10.15497348593496, 10.75576259882908, 11.25822540835418, 11.82352635463387, 12.19444746511934, 12.48693240162143, 13.20765239732595, 13.87354483112647, 14.07582899575711, 14.87399930440417
