| L(s) = 1 | + 2.23·2-s + 3.00·4-s + 2.92·5-s − 4.53·7-s + 2.23·8-s + 6.53·10-s + 2.92·11-s + 6.53·13-s − 10.1·14-s − 0.999·16-s − 5.84·17-s − 4.53·19-s + 8.76·20-s + 6.53·22-s + 1.36·23-s + 3.53·25-s + 14.6·26-s − 13.5·28-s − 8.76·29-s − 6.70·32-s − 13.0·34-s − 13.2·35-s + 6·37-s − 10.1·38-s + 6.53·40-s + 2.73·41-s + 43-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.50·4-s + 1.30·5-s − 1.71·7-s + 0.790·8-s + 2.06·10-s + 0.880·11-s + 1.81·13-s − 2.70·14-s − 0.249·16-s − 1.41·17-s − 1.03·19-s + 1.95·20-s + 1.39·22-s + 0.285·23-s + 0.706·25-s + 2.86·26-s − 2.56·28-s − 1.62·29-s − 1.18·32-s − 2.24·34-s − 2.23·35-s + 0.986·37-s − 1.64·38-s + 1.03·40-s + 0.427·41-s + 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.259374452\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.259374452\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 - 2.92T + 5T^{2} \) |
| 7 | \( 1 + 4.53T + 7T^{2} \) |
| 11 | \( 1 - 2.92T + 11T^{2} \) |
| 13 | \( 1 - 6.53T + 13T^{2} \) |
| 17 | \( 1 + 5.84T + 17T^{2} \) |
| 19 | \( 1 + 4.53T + 19T^{2} \) |
| 23 | \( 1 - 1.36T + 23T^{2} \) |
| 29 | \( 1 + 8.76T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 47 | \( 1 + 8.76T + 47T^{2} \) |
| 53 | \( 1 - 2.73T + 53T^{2} \) |
| 59 | \( 1 - 1.36T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 8.58T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 1.06T + 79T^{2} \) |
| 83 | \( 1 - 0.181T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 5.46T + 97T^{2} \) |
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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
−11.45567932700300576107504244372, −10.68157950568582602061923979671, −9.422346470952104269372607250688, −8.925972589806966707494382605762, −6.70926076337777882916617438547, −6.30288878782081730185262953034, −5.78276297610549388311913659704, −4.21320431105326447554644329596, −3.39096705200551199194777883530, −2.10403443827301470585073105971,
2.10403443827301470585073105971, 3.39096705200551199194777883530, 4.21320431105326447554644329596, 5.78276297610549388311913659704, 6.30288878782081730185262953034, 6.70926076337777882916617438547, 8.925972589806966707494382605762, 9.422346470952104269372607250688, 10.68157950568582602061923979671, 11.45567932700300576107504244372
