| L(s) = 1 | − 1.56·2-s + 0.438·4-s + 0.561·5-s + 2.43·8-s − 0.876·10-s + 2.56·11-s + 4.56·13-s − 4.68·16-s − 17-s + 7.68·19-s + 0.246·20-s − 4·22-s + 6.56·23-s − 4.68·25-s − 7.12·26-s − 8.24·29-s − 5.12·31-s + 2.43·32-s + 1.56·34-s + 3.12·37-s − 12·38-s + 1.36·40-s − 0.561·41-s − 7.68·43-s + 1.12·44-s − 10.2·46-s + 2.87·47-s + ⋯ |
| L(s) = 1 | − 1.10·2-s + 0.219·4-s + 0.251·5-s + 0.862·8-s − 0.277·10-s + 0.772·11-s + 1.26·13-s − 1.17·16-s − 0.242·17-s + 1.76·19-s + 0.0550·20-s − 0.852·22-s + 1.36·23-s − 0.936·25-s − 1.39·26-s − 1.53·29-s − 0.920·31-s + 0.431·32-s + 0.267·34-s + 0.513·37-s − 1.94·38-s + 0.216·40-s − 0.0876·41-s − 1.17·43-s + 0.169·44-s − 1.51·46-s + 0.419·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6964459117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6964459117\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 + 1.56T + 2T^{2} \) |
| 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 - 6.56T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 + 0.561T + 41T^{2} \) |
| 43 | \( 1 + 7.68T + 43T^{2} \) |
| 47 | \( 1 - 2.87T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 4.24T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 9.12T + 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 + 11.1T + 97T^{2} \) |
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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
−13.17881610070629439084152871695, −11.57920990313255823239008887407, −10.88544641750356637896804341168, −9.566903962123646453429318620819, −9.109039558567399799467043499439, −7.937675897743211811306160150671, −6.86926695009354832977930651402, −5.40651200462861411922059245438, −3.71785808773257501095602833558, −1.40112078647855694052798575627,
1.40112078647855694052798575627, 3.71785808773257501095602833558, 5.40651200462861411922059245438, 6.86926695009354832977930651402, 7.937675897743211811306160150671, 9.109039558567399799467043499439, 9.566903962123646453429318620819, 10.88544641750356637896804341168, 11.57920990313255823239008887407, 13.17881610070629439084152871695
