| L(s) = 1 | − 2-s + 2.07·3-s + 4-s + 0.701·5-s − 2.07·6-s + 3.38·7-s − 8-s + 1.29·9-s − 0.701·10-s + 3.84·11-s + 2.07·12-s − 4.26·13-s − 3.38·14-s + 1.45·15-s + 16-s − 6.75·17-s − 1.29·18-s + 7.87·19-s + 0.701·20-s + 7.02·21-s − 3.84·22-s + 1.63·23-s − 2.07·24-s − 4.50·25-s + 4.26·26-s − 3.52·27-s + 3.38·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.19·3-s + 0.5·4-s + 0.313·5-s − 0.846·6-s + 1.28·7-s − 0.353·8-s + 0.433·9-s − 0.221·10-s + 1.15·11-s + 0.598·12-s − 1.18·13-s − 0.905·14-s + 0.375·15-s + 0.250·16-s − 1.63·17-s − 0.306·18-s + 1.80·19-s + 0.156·20-s + 1.53·21-s − 0.819·22-s + 0.341·23-s − 0.423·24-s − 0.901·25-s + 0.836·26-s − 0.678·27-s + 0.640·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 446 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.725986085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.725986085\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 223 | \( 1 - T \) |
| good | 3 | \( 1 - 2.07T + 3T^{2} \) |
| 5 | \( 1 - 0.701T + 5T^{2} \) |
| 7 | \( 1 - 3.38T + 7T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 4.26T + 13T^{2} \) |
| 17 | \( 1 + 6.75T + 17T^{2} \) |
| 19 | \( 1 - 7.87T + 19T^{2} \) |
| 23 | \( 1 - 1.63T + 23T^{2} \) |
| 29 | \( 1 + 2.14T + 29T^{2} \) |
| 31 | \( 1 + 2.31T + 31T^{2} \) |
| 37 | \( 1 - 5.66T + 37T^{2} \) |
| 41 | \( 1 - 1.47T + 41T^{2} \) |
| 43 | \( 1 - 6.07T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 7.27T + 53T^{2} \) |
| 59 | \( 1 - 8.21T + 59T^{2} \) |
| 61 | \( 1 - 7.24T + 61T^{2} \) |
| 67 | \( 1 - 9.74T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 + 9.88T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 0.294T + 97T^{2} \) |
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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
−11.33999500894787073162563667583, −9.649892006916561622767165444629, −9.403857250297482090444638167529, −8.422312540218272778314212289433, −7.69699216364205051435935008561, −6.86306993331133802901660710884, −5.35219578825523089197536394566, −4.08418613294721157162008534273, −2.61440936880690905375854100277, −1.64498287147741973280141769488,
1.64498287147741973280141769488, 2.61440936880690905375854100277, 4.08418613294721157162008534273, 5.35219578825523089197536394566, 6.86306993331133802901660710884, 7.69699216364205051435935008561, 8.422312540218272778314212289433, 9.403857250297482090444638167529, 9.649892006916561622767165444629, 11.33999500894787073162563667583
