| L(s) = 1 | + 0.911·2-s − 0.648·3-s − 1.16·4-s − 1.23·5-s − 0.590·6-s − 1.76·7-s − 2.88·8-s − 2.57·9-s − 1.12·10-s + 0.758·12-s − 4.35·13-s − 1.60·14-s + 0.801·15-s − 0.292·16-s + 7.66·17-s − 2.35·18-s + 4.10·19-s + 1.44·20-s + 1.14·21-s + 8.58·23-s + 1.87·24-s − 3.47·25-s − 3.96·26-s + 3.61·27-s + 2.05·28-s + 8.70·29-s + 0.730·30-s + ⋯ |
| L(s) = 1 | + 0.644·2-s − 0.374·3-s − 0.584·4-s − 0.552·5-s − 0.241·6-s − 0.665·7-s − 1.02·8-s − 0.859·9-s − 0.356·10-s + 0.218·12-s − 1.20·13-s − 0.428·14-s + 0.206·15-s − 0.0731·16-s + 1.85·17-s − 0.554·18-s + 0.942·19-s + 0.323·20-s + 0.249·21-s + 1.78·23-s + 0.382·24-s − 0.694·25-s − 0.778·26-s + 0.696·27-s + 0.389·28-s + 1.61·29-s + 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.015423312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015423312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| good | 2 | \( 1 - 0.911T + 2T^{2} \) |
| 3 | \( 1 + 0.648T + 3T^{2} \) |
| 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 + 1.76T + 7T^{2} \) |
| 13 | \( 1 + 4.35T + 13T^{2} \) |
| 17 | \( 1 - 7.66T + 17T^{2} \) |
| 19 | \( 1 - 4.10T + 19T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 - 8.70T + 29T^{2} \) |
| 31 | \( 1 + 0.393T + 31T^{2} \) |
| 37 | \( 1 - 0.904T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 + 9.86T + 43T^{2} \) |
| 47 | \( 1 - 2.45T + 47T^{2} \) |
| 53 | \( 1 - 1.96T + 53T^{2} \) |
| 59 | \( 1 - 0.899T + 59T^{2} \) |
| 61 | \( 1 + 8.22T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 + 7.67T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 - 1.25T + 83T^{2} \) |
| 89 | \( 1 - 9.43T + 89T^{2} \) |
| 97 | \( 1 - 8.36T + 97T^{2} \) |
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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
−9.729515705904427292896096408538, −8.839649468232490619906543945164, −7.982243192876482861871209314848, −7.10526494411501166990127321390, −6.08334305270479418044645941056, −5.24110669229607608342840130934, −4.76447614733139140865596985450, −3.29609159952889556773873517047, −3.06418145566942095341380422630, −0.66662684776985552149344903136,
0.66662684776985552149344903136, 3.06418145566942095341380422630, 3.29609159952889556773873517047, 4.76447614733139140865596985450, 5.24110669229607608342840130934, 6.08334305270479418044645941056, 7.10526494411501166990127321390, 7.982243192876482861871209314848, 8.839649468232490619906543945164, 9.729515705904427292896096408538
