| L(s) = 1 | + 3.70·2-s + 3·3-s + 5.70·4-s − 5.05·5-s + 11.1·6-s − 7·7-s − 8.50·8-s + 9·9-s − 18.7·10-s − 31.4·11-s + 17.1·12-s − 59.0·13-s − 25.9·14-s − 15.1·15-s − 77.1·16-s − 116.·17-s + 33.3·18-s + 122.·19-s − 28.8·20-s − 21·21-s − 116.·22-s − 23·23-s − 25.5·24-s − 99.4·25-s − 218.·26-s + 27·27-s − 39.9·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.577·3-s + 0.712·4-s − 0.452·5-s + 0.755·6-s − 0.377·7-s − 0.375·8-s + 0.333·9-s − 0.591·10-s − 0.863·11-s + 0.411·12-s − 1.25·13-s − 0.494·14-s − 0.261·15-s − 1.20·16-s − 1.66·17-s + 0.436·18-s + 1.47·19-s − 0.322·20-s − 0.218·21-s − 1.12·22-s − 0.208·23-s − 0.217·24-s − 0.795·25-s − 1.64·26-s + 0.192·27-s − 0.269·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 3.70T + 8T^{2} \) |
| 5 | \( 1 + 5.05T + 125T^{2} \) |
| 11 | \( 1 + 31.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 59.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 48.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 405.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 62.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 376.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 277.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 478.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 64.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 456.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 230.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 372.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.16e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 794.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 579.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 983.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.38e3T + 9.12e5T^{2} \) |
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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
−10.06112578290459170321128590118, −9.321203975428876623450623243136, −8.148408946972322093130377675807, −7.25461993774901886206589840520, −6.25749778853553382206832395794, −5.01622764424536686654508928635, −4.37666341569694277220701977527, −3.16420591363827880420440490887, −2.42196336507189864477555674313, 0,
2.42196336507189864477555674313, 3.16420591363827880420440490887, 4.37666341569694277220701977527, 5.01622764424536686654508928635, 6.25749778853553382206832395794, 7.25461993774901886206589840520, 8.148408946972322093130377675807, 9.321203975428876623450623243136, 10.06112578290459170321128590118
