| L(s) = 1 | + 1.98·2-s + 3-s + 1.95·4-s − 1.78·5-s + 1.98·6-s − 0.840·7-s − 0.0814·8-s + 9-s − 3.55·10-s − 6.06·11-s + 1.95·12-s − 4.54·13-s − 1.67·14-s − 1.78·15-s − 4.08·16-s + 5.04·17-s + 1.98·18-s − 0.331·19-s − 3.50·20-s − 0.840·21-s − 12.0·22-s + 23-s − 0.0814·24-s − 1.80·25-s − 9.03·26-s + 27-s − 1.64·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s + 0.577·3-s + 0.979·4-s − 0.799·5-s + 0.812·6-s − 0.317·7-s − 0.0288·8-s + 0.333·9-s − 1.12·10-s − 1.83·11-s + 0.565·12-s − 1.25·13-s − 0.447·14-s − 0.461·15-s − 1.02·16-s + 1.22·17-s + 0.468·18-s − 0.0759·19-s − 0.782·20-s − 0.183·21-s − 2.57·22-s + 0.208·23-s − 0.0166·24-s − 0.361·25-s − 1.77·26-s + 0.192·27-s − 0.311·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 + 0.840T + 7T^{2} \) |
| 11 | \( 1 + 6.06T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 5.04T + 17T^{2} \) |
| 19 | \( 1 + 0.331T + 19T^{2} \) |
| 31 | \( 1 - 0.306T + 31T^{2} \) |
| 37 | \( 1 - 8.76T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 - 8.93T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 7.91T + 71T^{2} \) |
| 73 | \( 1 - 2.31T + 73T^{2} \) |
| 79 | \( 1 + 3.85T + 79T^{2} \) |
| 83 | \( 1 - 9.11T + 83T^{2} \) |
| 89 | \( 1 + 0.431T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−8.527412025595522429933423348187, −7.76325140606858243100741284892, −7.30748947167281556248848245981, −6.20778454784127851435522182577, −5.15263622121855022852395431148, −4.83069051743904971821740615058, −3.66984452968375463229237287443, −3.06911542463815757260558197807, −2.29396793374380473688411339290, 0,
2.29396793374380473688411339290, 3.06911542463815757260558197807, 3.66984452968375463229237287443, 4.83069051743904971821740615058, 5.15263622121855022852395431148, 6.20778454784127851435522182577, 7.30748947167281556248848245981, 7.76325140606858243100741284892, 8.527412025595522429933423348187
