| L(s) = 1 | + 2.46·2-s − 1.08·3-s + 4.07·4-s − 1.96·5-s − 2.68·6-s − 2.04·7-s + 5.12·8-s − 1.81·9-s − 4.84·10-s − 11-s − 4.43·12-s + 2.80·13-s − 5.03·14-s + 2.13·15-s + 4.47·16-s − 5.12·17-s − 4.47·18-s − 3.46·19-s − 8.01·20-s + 2.22·21-s − 2.46·22-s − 2.42·23-s − 5.57·24-s − 1.14·25-s + 6.90·26-s + 5.24·27-s − 8.32·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.628·3-s + 2.03·4-s − 0.878·5-s − 1.09·6-s − 0.771·7-s + 1.81·8-s − 0.604·9-s − 1.53·10-s − 0.301·11-s − 1.28·12-s + 0.776·13-s − 1.34·14-s + 0.552·15-s + 1.11·16-s − 1.24·17-s − 1.05·18-s − 0.794·19-s − 1.79·20-s + 0.485·21-s − 0.525·22-s − 0.506·23-s − 1.13·24-s − 0.228·25-s + 1.35·26-s + 1.00·27-s − 1.57·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 2.46T + 2T^{2} \) |
| 3 | \( 1 + 1.08T + 3T^{2} \) |
| 5 | \( 1 + 1.96T + 5T^{2} \) |
| 7 | \( 1 + 2.04T + 7T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 + 2.42T + 23T^{2} \) |
| 29 | \( 1 + 10.7T + 29T^{2} \) |
| 31 | \( 1 - 9.00T + 31T^{2} \) |
| 37 | \( 1 - 0.168T + 37T^{2} \) |
| 41 | \( 1 - 0.663T + 41T^{2} \) |
| 43 | \( 1 - 6.25T + 43T^{2} \) |
| 47 | \( 1 + 3.60T + 47T^{2} \) |
| 53 | \( 1 - 2.61T + 53T^{2} \) |
| 59 | \( 1 + 9.36T + 59T^{2} \) |
| 61 | \( 1 - 9.84T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 1.39T + 71T^{2} \) |
| 73 | \( 1 - 2.44T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 + 0.911T + 89T^{2} \) |
| 97 | \( 1 + 2.45T + 97T^{2} \) |
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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
−9.057530000694766066729187000801, −8.105317713794150451737149766684, −7.09309979462130121601240517550, −6.20515264933418494238722732686, −5.93688584476610720432991369102, −4.78570821746183616813667809875, −4.06748855741494438897425628768, −3.32149785197155564012327263101, −2.28164568730898879255365938999, 0,
2.28164568730898879255365938999, 3.32149785197155564012327263101, 4.06748855741494438897425628768, 4.78570821746183616813667809875, 5.93688584476610720432991369102, 6.20515264933418494238722732686, 7.09309979462130121601240517550, 8.105317713794150451737149766684, 9.057530000694766066729187000801
