| L(s) = 1 | − 2-s + 1.56·3-s + 4-s + 2.32·5-s − 1.56·6-s + 0.0710·7-s − 8-s − 0.559·9-s − 2.32·10-s + 1.39·11-s + 1.56·12-s + 4.74·13-s − 0.0710·14-s + 3.62·15-s + 16-s + 5.64·17-s + 0.559·18-s − 2.22·19-s + 2.32·20-s + 0.110·21-s − 1.39·22-s − 1.47·23-s − 1.56·24-s + 0.390·25-s − 4.74·26-s − 5.56·27-s + 0.0710·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.901·3-s + 0.5·4-s + 1.03·5-s − 0.637·6-s + 0.0268·7-s − 0.353·8-s − 0.186·9-s − 0.734·10-s + 0.421·11-s + 0.450·12-s + 1.31·13-s − 0.0189·14-s + 0.936·15-s + 0.250·16-s + 1.36·17-s + 0.131·18-s − 0.510·19-s + 0.519·20-s + 0.0242·21-s − 0.298·22-s − 0.306·23-s − 0.318·24-s + 0.0781·25-s − 0.931·26-s − 1.07·27-s + 0.0134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.795291734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.795291734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 5 | \( 1 - 2.32T + 5T^{2} \) |
| 7 | \( 1 - 0.0710T + 7T^{2} \) |
| 11 | \( 1 - 1.39T + 11T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 17 | \( 1 - 5.64T + 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 - 2.67T + 29T^{2} \) |
| 37 | \( 1 - 8.42T + 37T^{2} \) |
| 41 | \( 1 - 2.66T + 41T^{2} \) |
| 43 | \( 1 + 4.72T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 8.51T + 53T^{2} \) |
| 59 | \( 1 + 3.25T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 - 5.19T + 67T^{2} \) |
| 71 | \( 1 + 9.18T + 71T^{2} \) |
| 73 | \( 1 + 2.13T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 8.10T + 89T^{2} \) |
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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
−8.158844221377354220040079262013, −7.69704532235931845321336740516, −6.65462554205501662543134774832, −5.96699192140813472414973213612, −5.59516547734614374134249886705, −4.21926170994056985029775768080, −3.39731192666989330420278956394, −2.64627340275741552277462563352, −1.81593393819855399328374090832, −0.994184289305892965245359196969,
0.994184289305892965245359196969, 1.81593393819855399328374090832, 2.64627340275741552277462563352, 3.39731192666989330420278956394, 4.21926170994056985029775768080, 5.59516547734614374134249886705, 5.96699192140813472414973213612, 6.65462554205501662543134774832, 7.69704532235931845321336740516, 8.158844221377354220040079262013
