| L(s) = 1 | + 2-s − 0.803·3-s + 4-s − 5-s − 0.803·6-s + 1.44·7-s + 8-s − 2.35·9-s − 10-s − 0.326·11-s − 0.803·12-s − 2.95·13-s + 1.44·14-s + 0.803·15-s + 16-s + 3.19·17-s − 2.35·18-s − 2.71·19-s − 20-s − 1.16·21-s − 0.326·22-s + 7.05·23-s − 0.803·24-s + 25-s − 2.95·26-s + 4.30·27-s + 1.44·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.464·3-s + 0.5·4-s − 0.447·5-s − 0.328·6-s + 0.546·7-s + 0.353·8-s − 0.784·9-s − 0.316·10-s − 0.0983·11-s − 0.232·12-s − 0.818·13-s + 0.386·14-s + 0.207·15-s + 0.250·16-s + 0.774·17-s − 0.554·18-s − 0.623·19-s − 0.223·20-s − 0.253·21-s − 0.0695·22-s + 1.47·23-s − 0.164·24-s + 0.200·25-s − 0.578·26-s + 0.828·27-s + 0.273·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 0.803T + 3T^{2} \) |
| 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 + 0.326T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 + 2.71T + 19T^{2} \) |
| 23 | \( 1 - 7.05T + 23T^{2} \) |
| 29 | \( 1 + 7.21T + 29T^{2} \) |
| 31 | \( 1 + 3.42T + 31T^{2} \) |
| 37 | \( 1 - 6.36T + 37T^{2} \) |
| 41 | \( 1 + 0.336T + 41T^{2} \) |
| 43 | \( 1 + 4.62T + 43T^{2} \) |
| 47 | \( 1 + 8.73T + 47T^{2} \) |
| 53 | \( 1 + 2.27T + 53T^{2} \) |
| 59 | \( 1 + 4.52T + 59T^{2} \) |
| 61 | \( 1 - 2.03T + 61T^{2} \) |
| 67 | \( 1 + 0.347T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 5.02T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 2.99T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−7.894366033429978564309251842435, −7.33105431842258423514792501370, −6.50828126003135994720772639629, −5.66124047440048850296216326637, −5.08071316228706951231570094254, −4.47432655830265274832247274560, −3.40277971383177729790816414877, −2.70307577274699567633661052600, −1.50338848745475014967223559464, 0,
1.50338848745475014967223559464, 2.70307577274699567633661052600, 3.40277971383177729790816414877, 4.47432655830265274832247274560, 5.08071316228706951231570094254, 5.66124047440048850296216326637, 6.50828126003135994720772639629, 7.33105431842258423514792501370, 7.894366033429978564309251842435
