| L(s) = 1 | − 2-s + 2.72·3-s + 4-s + 3.66·5-s − 2.72·6-s + 2.25·7-s − 8-s + 4.41·9-s − 3.66·10-s + 1.84·11-s + 2.72·12-s − 6.09·13-s − 2.25·14-s + 9.97·15-s + 16-s + 1.75·17-s − 4.41·18-s + 19-s + 3.66·20-s + 6.13·21-s − 1.84·22-s + 4.20·23-s − 2.72·24-s + 8.41·25-s + 6.09·26-s + 3.86·27-s + 2.25·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s + 1.63·5-s − 1.11·6-s + 0.851·7-s − 0.353·8-s + 1.47·9-s − 1.15·10-s + 0.555·11-s + 0.786·12-s − 1.69·13-s − 0.602·14-s + 2.57·15-s + 0.250·16-s + 0.425·17-s − 1.04·18-s + 0.229·19-s + 0.819·20-s + 1.33·21-s − 0.392·22-s + 0.877·23-s − 0.555·24-s + 1.68·25-s + 1.19·26-s + 0.743·27-s + 0.425·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.478512204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.478512204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.72T + 3T^{2} \) |
| 5 | \( 1 - 3.66T + 5T^{2} \) |
| 7 | \( 1 - 2.25T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 + 6.09T + 13T^{2} \) |
| 17 | \( 1 - 1.75T + 17T^{2} \) |
| 23 | \( 1 - 4.20T + 23T^{2} \) |
| 29 | \( 1 + 2.32T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 3.99T + 37T^{2} \) |
| 41 | \( 1 - 8.49T + 41T^{2} \) |
| 43 | \( 1 - 1.92T + 43T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 + 4.67T + 53T^{2} \) |
| 59 | \( 1 + 6.85T + 59T^{2} \) |
| 61 | \( 1 + 3.07T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 + 9.11T + 71T^{2} \) |
| 73 | \( 1 - 4.73T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 7.50T + 83T^{2} \) |
| 89 | \( 1 - 5.63T + 89T^{2} \) |
| 97 | \( 1 + 5.02T + 97T^{2} \) |
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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
−7.82861238994346733424868507318, −7.45097398444789047013904549286, −6.69144330785025053709658410854, −5.83228481903478084862860748882, −5.01348080546006937268907468223, −4.29404394031450616498850011437, −2.90838226108142164628690803614, −2.63802046992821673736462990159, −1.81585527047050153573392593163, −1.20256509297197204630434490671,
1.20256509297197204630434490671, 1.81585527047050153573392593163, 2.63802046992821673736462990159, 2.90838226108142164628690803614, 4.29404394031450616498850011437, 5.01348080546006937268907468223, 5.83228481903478084862860748882, 6.69144330785025053709658410854, 7.45097398444789047013904549286, 7.82861238994346733424868507318
