| L(s) = 1 | + 2-s + 2.97·3-s + 4-s + 3.26·5-s + 2.97·6-s − 5.07·7-s + 8-s + 5.84·9-s + 3.26·10-s + 11-s + 2.97·12-s + 5.09·13-s − 5.07·14-s + 9.69·15-s + 16-s − 0.859·17-s + 5.84·18-s − 0.792·19-s + 3.26·20-s − 15.0·21-s + 22-s + 4.06·23-s + 2.97·24-s + 5.63·25-s + 5.09·26-s + 8.45·27-s − 5.07·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.71·3-s + 0.5·4-s + 1.45·5-s + 1.21·6-s − 1.91·7-s + 0.353·8-s + 1.94·9-s + 1.03·10-s + 0.301·11-s + 0.858·12-s + 1.41·13-s − 1.35·14-s + 2.50·15-s + 0.250·16-s − 0.208·17-s + 1.37·18-s − 0.181·19-s + 0.729·20-s − 3.29·21-s + 0.213·22-s + 0.847·23-s + 0.607·24-s + 1.12·25-s + 0.998·26-s + 1.62·27-s − 0.958·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.751562559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.751562559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 - 3.26T + 5T^{2} \) |
| 7 | \( 1 + 5.07T + 7T^{2} \) |
| 13 | \( 1 - 5.09T + 13T^{2} \) |
| 17 | \( 1 + 0.859T + 17T^{2} \) |
| 19 | \( 1 + 0.792T + 19T^{2} \) |
| 23 | \( 1 - 4.06T + 23T^{2} \) |
| 29 | \( 1 + 9.45T + 29T^{2} \) |
| 31 | \( 1 - 6.80T + 31T^{2} \) |
| 37 | \( 1 + 3.56T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 - 8.42T + 43T^{2} \) |
| 47 | \( 1 - 7.88T + 47T^{2} \) |
| 53 | \( 1 - 1.35T + 53T^{2} \) |
| 59 | \( 1 + 9.72T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 + 7.94T + 71T^{2} \) |
| 73 | \( 1 - 7.55T + 73T^{2} \) |
| 79 | \( 1 + 0.246T + 79T^{2} \) |
| 83 | \( 1 - 5.15T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.702424427066592002525432371496, −7.51342280625355809992119469676, −6.76319596541304147977594660322, −6.22043029454760957842811874896, −5.64221452613773082498915669220, −4.30603104589103433929026923690, −3.47165763685896639477299585175, −3.08303613493643302706719101619, −2.27229287159036800447210420461, −1.39808204886272793646701439962,
1.39808204886272793646701439962, 2.27229287159036800447210420461, 3.08303613493643302706719101619, 3.47165763685896639477299585175, 4.30603104589103433929026923690, 5.64221452613773082498915669220, 6.22043029454760957842811874896, 6.76319596541304147977594660322, 7.51342280625355809992119469676, 8.702424427066592002525432371496
