| L(s) = 1 | + 4.30·2-s − 3·3-s + 10.5·4-s − 16.7·5-s − 12.9·6-s + 7·7-s + 10.8·8-s + 9·9-s − 71.9·10-s − 29.7·11-s − 31.5·12-s + 74.3·13-s + 30.1·14-s + 50.1·15-s − 37.4·16-s + 125.·17-s + 38.7·18-s + 111.·19-s − 175.·20-s − 21·21-s − 127.·22-s + 23·23-s − 32.5·24-s + 154.·25-s + 320.·26-s − 27·27-s + 73.6·28-s + ⋯ |
| L(s) = 1 | + 1.52·2-s − 0.577·3-s + 1.31·4-s − 1.49·5-s − 0.878·6-s + 0.377·7-s + 0.479·8-s + 0.333·9-s − 2.27·10-s − 0.815·11-s − 0.759·12-s + 1.58·13-s + 0.575·14-s + 0.863·15-s − 0.585·16-s + 1.78·17-s + 0.507·18-s + 1.34·19-s − 1.96·20-s − 0.218·21-s − 1.24·22-s + 0.208·23-s − 0.276·24-s + 1.23·25-s + 2.41·26-s − 0.192·27-s + 0.497·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.098893347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.098893347\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 4.30T + 8T^{2} \) |
| 5 | \( 1 + 16.7T + 125T^{2} \) |
| 11 | \( 1 + 29.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 74.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 125.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 221.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 203.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 391.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 317.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 329.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 631.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 153.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 767.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.14e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 225.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 437.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 709.T + 9.12e5T^{2} \) |
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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
−11.13186043312071389434331507387, −10.05596464765254163185616820900, −8.364760956217258157924008765323, −7.70553124816745066306363805351, −6.62400670048703290698013972738, −5.52556251227177186746226359213, −4.87974384393031305317773602789, −3.75330538895241834487226866303, −3.16582194496744012674785434392, −0.945080654925863133901101849933,
0.945080654925863133901101849933, 3.16582194496744012674785434392, 3.75330538895241834487226866303, 4.87974384393031305317773602789, 5.52556251227177186746226359213, 6.62400670048703290698013972738, 7.70553124816745066306363805351, 8.364760956217258157924008765323, 10.05596464765254163185616820900, 11.13186043312071389434331507387
