| L(s) = 1 | − 2-s − 3.08·3-s + 4-s − 2.32·5-s + 3.08·6-s − 8-s + 6.49·9-s + 2.32·10-s + 4.60·11-s − 3.08·12-s − 6.25·13-s + 7.15·15-s + 16-s − 4.31·17-s − 6.49·18-s + 8.07·19-s − 2.32·20-s − 4.60·22-s − 2.30·23-s + 3.08·24-s + 0.382·25-s + 6.25·26-s − 10.7·27-s − 7.84·29-s − 7.15·30-s − 4.09·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.77·3-s + 0.5·4-s − 1.03·5-s + 1.25·6-s − 0.353·8-s + 2.16·9-s + 0.733·10-s + 1.38·11-s − 0.889·12-s − 1.73·13-s + 1.84·15-s + 0.250·16-s − 1.04·17-s − 1.53·18-s + 1.85·19-s − 0.518·20-s − 0.982·22-s − 0.479·23-s + 0.629·24-s + 0.0765·25-s + 1.22·26-s − 2.07·27-s − 1.45·29-s − 1.30·30-s − 0.736·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 + 3.08T + 3T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 11 | \( 1 - 4.60T + 11T^{2} \) |
| 13 | \( 1 + 6.25T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 - 8.07T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 + 7.84T + 29T^{2} \) |
| 31 | \( 1 + 4.09T + 31T^{2} \) |
| 37 | \( 1 + 1.22T + 37T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 7.51T + 53T^{2} \) |
| 59 | \( 1 - 2.94T + 59T^{2} \) |
| 61 | \( 1 - 6.03T + 61T^{2} \) |
| 67 | \( 1 + 5.05T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 0.348T + 79T^{2} \) |
| 83 | \( 1 - 2.13T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 - 12.6T + 97T^{2} \) |
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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
−7.69583656028894736910898438989, −7.27604429472552894262082233149, −6.85772810211332387234967994519, −5.86458539069556391995894393352, −5.23631005916050358004455276581, −4.33395603633699699184927650640, −3.68223877476602211427404172488, −2.10270942713587115919587226301, −0.892348448236604289493500939236, 0,
0.892348448236604289493500939236, 2.10270942713587115919587226301, 3.68223877476602211427404172488, 4.33395603633699699184927650640, 5.23631005916050358004455276581, 5.86458539069556391995894393352, 6.85772810211332387234967994519, 7.27604429472552894262082233149, 7.69583656028894736910898438989
