| L(s) = 1 | − 2.59·2-s − 3-s + 4.73·4-s + 2.59·6-s + 3.93·7-s − 7.10·8-s + 9-s + 2.24·11-s − 4.73·12-s − 2.09·13-s − 10.2·14-s + 8.97·16-s − 6.95·17-s − 2.59·18-s + 4.29·19-s − 3.93·21-s − 5.83·22-s + 5.19·23-s + 7.10·24-s + 5.43·26-s − 27-s + 18.6·28-s + 29-s + 2.25·31-s − 9.08·32-s − 2.24·33-s + 18.0·34-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 0.577·3-s + 2.36·4-s + 1.05·6-s + 1.48·7-s − 2.51·8-s + 0.333·9-s + 0.677·11-s − 1.36·12-s − 0.580·13-s − 2.73·14-s + 2.24·16-s − 1.68·17-s − 0.611·18-s + 0.985·19-s − 0.859·21-s − 1.24·22-s + 1.08·23-s + 1.45·24-s + 1.06·26-s − 0.192·27-s + 3.52·28-s + 0.185·29-s + 0.404·31-s − 1.60·32-s − 0.391·33-s + 3.09·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7465838092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7465838092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.59T + 2T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 17 | \( 1 + 6.95T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 - 5.19T + 23T^{2} \) |
| 31 | \( 1 - 2.25T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 2.66T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 - 7.74T + 47T^{2} \) |
| 53 | \( 1 - 1.41T + 53T^{2} \) |
| 59 | \( 1 - 5.76T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + 4.55T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 3.81T + 83T^{2} \) |
| 89 | \( 1 + 2.07T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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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
−9.054443128455907529195349297512, −8.405350027428219316928162443420, −7.63022892143888121680143117719, −7.02441396769026532923021784992, −6.32094479800245635144797047903, −5.17603007417575509697511783192, −4.35844636948417491593736125408, −2.65670989075033562603567888800, −1.69678880257147556323904755680, −0.807133543343497993372000165724,
0.807133543343497993372000165724, 1.69678880257147556323904755680, 2.65670989075033562603567888800, 4.35844636948417491593736125408, 5.17603007417575509697511783192, 6.32094479800245635144797047903, 7.02441396769026532923021784992, 7.63022892143888121680143117719, 8.405350027428219316928162443420, 9.054443128455907529195349297512
