| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 4.61·7-s + 8-s + 9-s − 10-s − 5.10·11-s − 12-s − 3.23·13-s + 4.61·14-s + 15-s + 16-s + 18-s − 4·19-s − 20-s − 4.61·21-s − 5.10·22-s + 6.22·23-s − 24-s + 25-s − 3.23·26-s − 27-s + 4.61·28-s − 8.33·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.74·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s − 1.54·11-s − 0.288·12-s − 0.897·13-s + 1.23·14-s + 0.258·15-s + 0.250·16-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 1.00·21-s − 1.08·22-s + 1.29·23-s − 0.204·24-s + 0.200·25-s − 0.634·26-s − 0.192·27-s + 0.871·28-s − 1.54·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 - 4.61T + 7T^{2} \) |
| 11 | \( 1 + 5.10T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 6.22T + 23T^{2} \) |
| 29 | \( 1 + 8.33T + 29T^{2} \) |
| 31 | \( 1 - 3.79T + 31T^{2} \) |
| 37 | \( 1 - 7.49T + 37T^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 + 8.25T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 1.36T + 53T^{2} \) |
| 59 | \( 1 - 2.37T + 59T^{2} \) |
| 61 | \( 1 - 1.22T + 61T^{2} \) |
| 67 | \( 1 - 2.74T + 67T^{2} \) |
| 71 | \( 1 + 9.09T + 71T^{2} \) |
| 73 | \( 1 + 9.20T + 73T^{2} \) |
| 79 | \( 1 + 3.61T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 9.03T + 89T^{2} \) |
| 97 | \( 1 + 6.00T + 97T^{2} \) |
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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
−7.39967109634201745446957107169, −6.82687723090808621142351792289, −5.66815563674238473376815745253, −5.31920513754272275329594267908, −4.55381179827954284048552906284, −4.36668861894412942076975038308, −3.00021360742878582405221401806, −2.29760315636444016590524217407, −1.37745674648870847225892211535, 0,
1.37745674648870847225892211535, 2.29760315636444016590524217407, 3.00021360742878582405221401806, 4.36668861894412942076975038308, 4.55381179827954284048552906284, 5.31920513754272275329594267908, 5.66815563674238473376815745253, 6.82687723090808621142351792289, 7.39967109634201745446957107169
