| L(s) = 1 | − 1.09·2-s + 0.379·3-s − 0.795·4-s + 5-s − 0.416·6-s + 1.89·7-s + 3.06·8-s − 2.85·9-s − 1.09·10-s + 0.134·11-s − 0.301·12-s − 3.51·13-s − 2.07·14-s + 0.379·15-s − 1.77·16-s − 1.66·17-s + 3.13·18-s − 0.795·20-s + 0.718·21-s − 0.147·22-s + 5.36·23-s + 1.16·24-s + 25-s + 3.85·26-s − 2.22·27-s − 1.50·28-s − 4.97·29-s + ⋯ |
| L(s) = 1 | − 0.776·2-s + 0.219·3-s − 0.397·4-s + 0.447·5-s − 0.169·6-s + 0.715·7-s + 1.08·8-s − 0.952·9-s − 0.347·10-s + 0.0405·11-s − 0.0871·12-s − 0.974·13-s − 0.555·14-s + 0.0979·15-s − 0.443·16-s − 0.402·17-s + 0.738·18-s − 0.177·20-s + 0.156·21-s − 0.0314·22-s + 1.11·23-s + 0.237·24-s + 0.200·25-s + 0.756·26-s − 0.427·27-s − 0.284·28-s − 0.923·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.09T + 2T^{2} \) |
| 3 | \( 1 - 0.379T + 3T^{2} \) |
| 7 | \( 1 - 1.89T + 7T^{2} \) |
| 11 | \( 1 - 0.134T + 11T^{2} \) |
| 13 | \( 1 + 3.51T + 13T^{2} \) |
| 17 | \( 1 + 1.66T + 17T^{2} \) |
| 23 | \( 1 - 5.36T + 23T^{2} \) |
| 29 | \( 1 + 4.97T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 - 1.69T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 8.50T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 - 0.264T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 - 9.17T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 + 1.32T + 71T^{2} \) |
| 73 | \( 1 + 6.34T + 73T^{2} \) |
| 79 | \( 1 - 1.46T + 79T^{2} \) |
| 83 | \( 1 - 7.44T + 83T^{2} \) |
| 89 | \( 1 + 9.73T + 89T^{2} \) |
| 97 | \( 1 + 17.4T + 97T^{2} \) |
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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
−8.946845998585053007737757440696, −8.252058149457871605120949218202, −7.55139482066445965199876792645, −6.71949400603587628508691324235, −5.35808491183678254085268779575, −5.02479799624286547433525875574, −3.82395968936054226756220164693, −2.55784431112748628696764641358, −1.55021993786131977251769176065, 0,
1.55021993786131977251769176065, 2.55784431112748628696764641358, 3.82395968936054226756220164693, 5.02479799624286547433525875574, 5.35808491183678254085268779575, 6.71949400603587628508691324235, 7.55139482066445965199876792645, 8.252058149457871605120949218202, 8.946845998585053007737757440696
