| L(s) = 1 | − 4.61·2-s − 3·3-s + 13.2·4-s + 17.5·5-s + 13.8·6-s − 13.8·7-s − 24.4·8-s + 9·9-s − 80.7·10-s − 71.1·11-s − 39.8·12-s − 33.5·13-s + 63.7·14-s − 52.5·15-s + 6.28·16-s − 109.·17-s − 41.5·18-s + 59.1·19-s + 232.·20-s + 41.4·21-s + 328.·22-s − 6.21·23-s + 73.2·24-s + 181.·25-s + 154.·26-s − 27·27-s − 183.·28-s + ⋯ |
| L(s) = 1 | − 1.63·2-s − 0.577·3-s + 1.66·4-s + 1.56·5-s + 0.941·6-s − 0.746·7-s − 1.07·8-s + 0.333·9-s − 2.55·10-s − 1.95·11-s − 0.959·12-s − 0.716·13-s + 1.21·14-s − 0.904·15-s + 0.0982·16-s − 1.55·17-s − 0.543·18-s + 0.713·19-s + 2.60·20-s + 0.430·21-s + 3.18·22-s − 0.0563·23-s + 0.622·24-s + 1.45·25-s + 1.16·26-s − 0.192·27-s − 1.23·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 723 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4631991564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4631991564\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 241 | \( 1 + 241T \) |
| good | 2 | \( 1 + 4.61T + 8T^{2} \) |
| 5 | \( 1 - 17.5T + 125T^{2} \) |
| 7 | \( 1 + 13.8T + 343T^{2} \) |
| 11 | \( 1 + 71.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 33.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 59.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + 6.21T + 1.21e4T^{2} \) |
| 29 | \( 1 - 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 308.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 251.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 218.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 244.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 639.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 394.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 257.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 381.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 488.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 703.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 707.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 289.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 83.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 260.T + 9.12e5T^{2} \) |
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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
−9.894515458463997327206303498580, −9.438841444921433077647417078273, −8.512601874463823288855527754938, −7.39357122375460143085620854817, −6.71625400145452838138324066607, −5.76539202136043756500640040648, −4.93962504185411979441918313999, −2.66076528332065240248743349525, −2.02464864048902934185389153215, −0.48362995601796702947720478371,
0.48362995601796702947720478371, 2.02464864048902934185389153215, 2.66076528332065240248743349525, 4.93962504185411979441918313999, 5.76539202136043756500640040648, 6.71625400145452838138324066607, 7.39357122375460143085620854817, 8.512601874463823288855527754938, 9.438841444921433077647417078273, 9.894515458463997327206303498580
