| L(s) = 1 | + 0.765·2-s + 3-s − 1.41·4-s + 0.0823·5-s + 0.765·6-s − 2.76·7-s − 2.61·8-s + 9-s + 0.0630·10-s − 0.668·11-s − 1.41·12-s − 3.28·13-s − 2.11·14-s + 0.0823·15-s + 0.828·16-s + 0.765·18-s + 3.64·19-s − 0.116·20-s − 2.76·21-s − 0.511·22-s − 9.30·23-s − 2.61·24-s − 4.99·25-s − 2.51·26-s + 27-s + 3.91·28-s − 6.24·29-s + ⋯ |
| L(s) = 1 | + 0.541·2-s + 0.577·3-s − 0.707·4-s + 0.0368·5-s + 0.312·6-s − 1.04·7-s − 0.923·8-s + 0.333·9-s + 0.0199·10-s − 0.201·11-s − 0.408·12-s − 0.910·13-s − 0.565·14-s + 0.0212·15-s + 0.207·16-s + 0.180·18-s + 0.835·19-s − 0.0260·20-s − 0.603·21-s − 0.109·22-s − 1.94·23-s − 0.533·24-s − 0.998·25-s − 0.492·26-s + 0.192·27-s + 0.739·28-s − 1.15·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.765T + 2T^{2} \) |
| 5 | \( 1 - 0.0823T + 5T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 + 0.668T + 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 19 | \( 1 - 3.64T + 19T^{2} \) |
| 23 | \( 1 + 9.30T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 + 5.04T + 31T^{2} \) |
| 37 | \( 1 + 2.40T + 37T^{2} \) |
| 41 | \( 1 - 0.480T + 41T^{2} \) |
| 43 | \( 1 - 8.27T + 43T^{2} \) |
| 47 | \( 1 + 8.88T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 9.59T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 0.944T + 67T^{2} \) |
| 71 | \( 1 - 3.28T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 8.37T + 79T^{2} \) |
| 83 | \( 1 - 0.899T + 83T^{2} \) |
| 89 | \( 1 - 5.64T + 89T^{2} \) |
| 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
−9.703766931908428812076985882971, −9.100701719497635542874776313839, −8.021926555524374426655054812913, −7.25212660308827933121795031778, −6.03386340767076854659317610013, −5.32430448440947130676770502512, −4.06709772566297443622278676198, −3.46076222304891927892173669255, −2.27388230644167949692486693486, 0,
2.27388230644167949692486693486, 3.46076222304891927892173669255, 4.06709772566297443622278676198, 5.32430448440947130676770502512, 6.03386340767076854659317610013, 7.25212660308827933121795031778, 8.021926555524374426655054812913, 9.100701719497635542874776313839, 9.703766931908428812076985882971
