| L(s) = 1 | + 2-s − 2.38·3-s + 4-s − 3.46·5-s − 2.38·6-s + 0.0815·7-s + 8-s + 2.69·9-s − 3.46·10-s − 11-s − 2.38·12-s + 3.82·13-s + 0.0815·14-s + 8.25·15-s + 16-s + 3.10·17-s + 2.69·18-s − 0.518·19-s − 3.46·20-s − 0.194·21-s − 22-s − 5.34·23-s − 2.38·24-s + 6.98·25-s + 3.82·26-s + 0.732·27-s + 0.0815·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.37·3-s + 0.5·4-s − 1.54·5-s − 0.974·6-s + 0.0308·7-s + 0.353·8-s + 0.897·9-s − 1.09·10-s − 0.301·11-s − 0.688·12-s + 1.06·13-s + 0.0218·14-s + 2.13·15-s + 0.250·16-s + 0.752·17-s + 0.634·18-s − 0.118·19-s − 0.774·20-s − 0.0424·21-s − 0.213·22-s − 1.11·23-s − 0.487·24-s + 1.39·25-s + 0.749·26-s + 0.141·27-s + 0.0154·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8764112537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8764112537\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.38T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 - 0.0815T + 7T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 - 3.10T + 17T^{2} \) |
| 19 | \( 1 + 0.518T + 19T^{2} \) |
| 23 | \( 1 + 5.34T + 23T^{2} \) |
| 29 | \( 1 + 9.87T + 29T^{2} \) |
| 31 | \( 1 + 7.74T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 + 8.22T + 41T^{2} \) |
| 43 | \( 1 - 5.61T + 43T^{2} \) |
| 47 | \( 1 - 4.71T + 47T^{2} \) |
| 53 | \( 1 - 5.19T + 53T^{2} \) |
| 59 | \( 1 + 9.19T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 2.10T + 71T^{2} \) |
| 73 | \( 1 + 6.06T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 3.11T + 89T^{2} \) |
| 97 | \( 1 - 7.84T + 97T^{2} \) |
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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
−8.093395887603499237655067058358, −7.49351919439380834722681792582, −6.84173788433030654737378614872, −5.95031789821033387488941098495, −5.46286844167193627199438350715, −4.71080788086490955687285461222, −3.67752578206066497015640791680, −3.57993099043377217001472337265, −1.83843387353609465999110649426, −0.50191359467835359011359927418,
0.50191359467835359011359927418, 1.83843387353609465999110649426, 3.57993099043377217001472337265, 3.67752578206066497015640791680, 4.71080788086490955687285461222, 5.46286844167193627199438350715, 5.95031789821033387488941098495, 6.84173788433030654737378614872, 7.49351919439380834722681792582, 8.093395887603499237655067058358
