| L(s) = 1 | + 0.414·2-s − 9.24·3-s − 7.82·4-s + 0.656·5-s − 3.82·6-s + 6.14·7-s − 6.55·8-s + 58.4·9-s + 0.272·10-s − 65.3·11-s + 72.3·12-s − 49.7·13-s + 2.54·14-s − 6.07·15-s + 59.9·16-s + 55.4·17-s + 24.2·18-s − 64.7·19-s − 5.14·20-s − 56.7·21-s − 27.0·22-s + 93.8·23-s + 60.5·24-s − 124.·25-s − 20.6·26-s − 290.·27-s − 48.0·28-s + ⋯ |
| L(s) = 1 | + 0.146·2-s − 1.77·3-s − 0.978·4-s + 0.0587·5-s − 0.260·6-s + 0.331·7-s − 0.289·8-s + 2.16·9-s + 0.00860·10-s − 1.79·11-s + 1.74·12-s − 1.06·13-s + 0.0485·14-s − 0.104·15-s + 0.936·16-s + 0.791·17-s + 0.316·18-s − 0.781·19-s − 0.0574·20-s − 0.589·21-s − 0.262·22-s + 0.851·23-s + 0.515·24-s − 0.996·25-s − 0.155·26-s − 2.07·27-s − 0.324·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 - 29T \) |
| good | 2 | \( 1 - 0.414T + 8T^{2} \) |
| 3 | \( 1 + 9.24T + 27T^{2} \) |
| 5 | \( 1 - 0.656T + 125T^{2} \) |
| 7 | \( 1 - 6.14T + 343T^{2} \) |
| 11 | \( 1 + 65.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 55.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 64.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 93.8T + 1.21e4T^{2} \) |
| 31 | \( 1 + 236.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 76.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 215.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 80.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 357.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 328.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 99.2T + 2.05e5T^{2} \) |
| 61 | \( 1 + 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 844.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 378.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 581.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 353.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 696.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 805.T + 9.12e5T^{2} \) |
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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
−16.33833750590260599028789669791, −14.90157465963589865550929850725, −13.12577067531635209909531596123, −12.36325042905113069824494992125, −10.89075172801174481768107751876, −9.862476516189287789103102992686, −7.66321302669574866883891850884, −5.62777663248966123170236433844, −4.79240736397582967434846963710, 0,
4.79240736397582967434846963710, 5.62777663248966123170236433844, 7.66321302669574866883891850884, 9.862476516189287789103102992686, 10.89075172801174481768107751876, 12.36325042905113069824494992125, 13.12577067531635209909531596123, 14.90157465963589865550929850725, 16.33833750590260599028789669791
