| L(s) = 1 | − 2.42·2-s − 1.02·3-s + 3.87·4-s − 0.275·5-s + 2.47·6-s − 5.16·7-s − 4.55·8-s − 1.95·9-s + 0.667·10-s − 4.59·11-s − 3.95·12-s + 2.58·13-s + 12.5·14-s + 0.280·15-s + 3.27·16-s + 6.77·17-s + 4.74·18-s − 7.93·19-s − 1.06·20-s + 5.26·21-s + 11.1·22-s + 23-s + 4.64·24-s − 4.92·25-s − 6.26·26-s + 5.05·27-s − 20.0·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 0.589·3-s + 1.93·4-s − 0.123·5-s + 1.00·6-s − 1.95·7-s − 1.60·8-s − 0.652·9-s + 0.210·10-s − 1.38·11-s − 1.14·12-s + 0.716·13-s + 3.34·14-s + 0.0724·15-s + 0.819·16-s + 1.64·17-s + 1.11·18-s − 1.82·19-s − 0.238·20-s + 1.14·21-s + 2.37·22-s + 0.208·23-s + 0.947·24-s − 0.984·25-s − 1.22·26-s + 0.973·27-s − 3.78·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1821784224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1821784224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.42T + 2T^{2} \) |
| 3 | \( 1 + 1.02T + 3T^{2} \) |
| 5 | \( 1 + 0.275T + 5T^{2} \) |
| 7 | \( 1 + 5.16T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 + 7.93T + 19T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 + 1.24T + 37T^{2} \) |
| 41 | \( 1 + 1.39T + 41T^{2} \) |
| 43 | \( 1 + 0.777T + 43T^{2} \) |
| 47 | \( 1 + 5.57T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 - 7.31T + 59T^{2} \) |
| 61 | \( 1 - 1.28T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 4.16T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 + 1.19T + 79T^{2} \) |
| 83 | \( 1 + 5.53T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 3.41T + 97T^{2} \) |
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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
−10.27192885364087112836193355731, −9.849993398421548375378008406572, −8.777088699758657053842709271638, −8.137787446727861726263591175485, −7.11300461458905456132862007958, −6.24812502115521984718628661676, −5.60356060324854214781198284705, −3.52170724335334249567494684274, −2.46059151784143982711822032450, −0.44665184182056498038474904150,
0.44665184182056498038474904150, 2.46059151784143982711822032450, 3.52170724335334249567494684274, 5.60356060324854214781198284705, 6.24812502115521984718628661676, 7.11300461458905456132862007958, 8.137787446727861726263591175485, 8.777088699758657053842709271638, 9.849993398421548375378008406572, 10.27192885364087112836193355731
