| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 3.37·7-s + 8-s + 9-s + 10-s + 11-s + 12-s + 4.74·13-s + 3.37·14-s + 15-s + 16-s + 17-s + 18-s + 4·19-s + 20-s + 3.37·21-s + 22-s − 3.37·23-s + 24-s + 25-s + 4.74·26-s + 27-s + 3.37·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s + 1.27·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.301·11-s + 0.288·12-s + 1.31·13-s + 0.901·14-s + 0.258·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s + 0.735·21-s + 0.213·22-s − 0.703·23-s + 0.204·24-s + 0.200·25-s + 0.930·26-s + 0.192·27-s + 0.637·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.809086434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.809086434\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 3.37T + 7T^{2} \) |
| 13 | \( 1 - 4.74T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 3.37T + 23T^{2} \) |
| 29 | \( 1 - 2.62T + 29T^{2} \) |
| 31 | \( 1 - 4.62T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 12.7T + 41T^{2} \) |
| 43 | \( 1 - 7.37T + 43T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 + 0.744T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 1.25T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 + 9.37T + 97T^{2} \) |
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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
−8.299900316175089647572466110095, −7.42812289881646403344433528819, −6.63291035737035358774974795939, −5.92568207832459889250767163038, −5.13242157497669802015675125328, −4.53777329724031084095520610088, −3.62415153635194243712689392677, −2.99486759739824171108882606540, −1.75888445558378289494918619386, −1.37188194691573686285945082571,
1.37188194691573686285945082571, 1.75888445558378289494918619386, 2.99486759739824171108882606540, 3.62415153635194243712689392677, 4.53777329724031084095520610088, 5.13242157497669802015675125328, 5.92568207832459889250767163038, 6.63291035737035358774974795939, 7.42812289881646403344433528819, 8.299900316175089647572466110095
