L(s) = 1 | − 2-s + 0.372·3-s + 4-s − 2.66·5-s − 0.372·6-s + 3.34·7-s − 8-s − 2.86·9-s + 2.66·10-s − 3.14·11-s + 0.372·12-s + 0.642·13-s − 3.34·14-s − 0.994·15-s + 16-s − 3.11·17-s + 2.86·18-s − 0.263·19-s − 2.66·20-s + 1.24·21-s + 3.14·22-s − 23-s − 0.372·24-s + 2.11·25-s − 0.642·26-s − 2.18·27-s + 3.34·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.215·3-s + 0.5·4-s − 1.19·5-s − 0.152·6-s + 1.26·7-s − 0.353·8-s − 0.953·9-s + 0.843·10-s − 0.949·11-s + 0.107·12-s + 0.178·13-s − 0.895·14-s − 0.256·15-s + 0.250·16-s − 0.755·17-s + 0.674·18-s − 0.0605·19-s − 0.596·20-s + 0.272·21-s + 0.671·22-s − 0.208·23-s − 0.0761·24-s + 0.422·25-s − 0.126·26-s − 0.420·27-s + 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8031032656\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8031032656\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 0.372T + 3T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 7 | \( 1 - 3.34T + 7T^{2} \) |
| 11 | \( 1 + 3.14T + 11T^{2} \) |
| 13 | \( 1 - 0.642T + 13T^{2} \) |
| 17 | \( 1 + 3.11T + 17T^{2} \) |
| 19 | \( 1 + 0.263T + 19T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 - 8.42T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 + 1.76T + 43T^{2} \) |
| 47 | \( 1 + 0.148T + 47T^{2} \) |
| 53 | \( 1 + 9.49T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 - 5.20T + 61T^{2} \) |
| 67 | \( 1 - 5.80T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 4.25T + 73T^{2} \) |
| 79 | \( 1 - 5.78T + 79T^{2} \) |
| 83 | \( 1 - 2.76T + 83T^{2} \) |
| 89 | \( 1 - 12.7T + 89T^{2} \) |
| 97 | \( 1 + 3.82T + 97T^{2} \) |
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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
−8.243970473492114319001552074141, −7.72687510567704107967837992646, −6.88838162919415615372858398374, −6.07260760021594547458395643141, −5.00785863837880112005413348343, −4.59487587028053494397476262865, −3.46799185767094723207280516038, −2.71743540006371112577854472153, −1.80525050302847172432254753498, −0.50444458741108891140563693857,
0.50444458741108891140563693857, 1.80525050302847172432254753498, 2.71743540006371112577854472153, 3.46799185767094723207280516038, 4.59487587028053494397476262865, 5.00785863837880112005413348343, 6.07260760021594547458395643141, 6.88838162919415615372858398374, 7.72687510567704107967837992646, 8.243970473492114319001552074141