| L(s) = 1 | + 1.55·2-s + 0.403·4-s − 1.95·5-s − 0.219·7-s − 2.47·8-s − 3.03·10-s − 0.381·11-s − 0.251·13-s − 0.340·14-s − 4.64·16-s − 5.53·19-s − 0.790·20-s − 0.590·22-s − 0.466·23-s − 1.16·25-s − 0.390·26-s − 0.0887·28-s − 3.49·29-s − 0.155·31-s − 2.25·32-s + 0.429·35-s + 9.10·37-s − 8.57·38-s + 4.84·40-s + 6.55·41-s + 7.56·43-s − 0.153·44-s + ⋯ |
| L(s) = 1 | + 1.09·2-s + 0.201·4-s − 0.875·5-s − 0.0830·7-s − 0.874·8-s − 0.959·10-s − 0.114·11-s − 0.0697·13-s − 0.0910·14-s − 1.16·16-s − 1.26·19-s − 0.176·20-s − 0.125·22-s − 0.0972·23-s − 0.233·25-s − 0.0764·26-s − 0.0167·28-s − 0.648·29-s − 0.0279·31-s − 0.398·32-s + 0.0726·35-s + 1.49·37-s − 1.39·38-s + 0.765·40-s + 1.02·41-s + 1.15·43-s − 0.0232·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664819441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664819441\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 5 | \( 1 + 1.95T + 5T^{2} \) |
| 7 | \( 1 + 0.219T + 7T^{2} \) |
| 11 | \( 1 + 0.381T + 11T^{2} \) |
| 13 | \( 1 + 0.251T + 13T^{2} \) |
| 19 | \( 1 + 5.53T + 19T^{2} \) |
| 23 | \( 1 + 0.466T + 23T^{2} \) |
| 29 | \( 1 + 3.49T + 29T^{2} \) |
| 31 | \( 1 + 0.155T + 31T^{2} \) |
| 37 | \( 1 - 9.10T + 37T^{2} \) |
| 41 | \( 1 - 6.55T + 41T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 - 5.19T + 47T^{2} \) |
| 53 | \( 1 + 6.96T + 53T^{2} \) |
| 59 | \( 1 + 1.15T + 59T^{2} \) |
| 61 | \( 1 - 2.51T + 61T^{2} \) |
| 67 | \( 1 + 9.92T + 67T^{2} \) |
| 71 | \( 1 - 9.29T + 71T^{2} \) |
| 73 | \( 1 + 6.94T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 + 1.48T + 83T^{2} \) |
| 89 | \( 1 - 1.72T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−7.79570273476716363340116712072, −7.11300068435997735807368124492, −6.13010778708071435758160498372, −5.87033328548893872001988281824, −4.77142287511627023612936644921, −4.31210797798314084192527761368, −3.75754911813318210467412739666, −2.94032307871489461923138489545, −2.10782351121796248837253563343, −0.51833172783843970652803372759,
0.51833172783843970652803372759, 2.10782351121796248837253563343, 2.94032307871489461923138489545, 3.75754911813318210467412739666, 4.31210797798314084192527761368, 4.77142287511627023612936644921, 5.87033328548893872001988281824, 6.13010778708071435758160498372, 7.11300068435997735807368124492, 7.79570273476716363340116712072
