| L(s) = 1 | + 2·2-s + 2·4-s + 5-s + 2·10-s + 3·11-s + 13-s − 4·16-s + 3·17-s + 19-s + 2·20-s + 6·22-s − 4·23-s + 25-s + 2·26-s − 5·29-s + 8·31-s − 8·32-s + 6·34-s − 12·37-s + 2·38-s − 8·41-s + 4·43-s + 6·44-s − 8·46-s − 7·47-s + 2·50-s + 2·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s + 0.632·10-s + 0.904·11-s + 0.277·13-s − 16-s + 0.727·17-s + 0.229·19-s + 0.447·20-s + 1.27·22-s − 0.834·23-s + 1/5·25-s + 0.392·26-s − 0.928·29-s + 1.43·31-s − 1.41·32-s + 1.02·34-s − 1.97·37-s + 0.324·38-s − 1.24·41-s + 0.609·43-s + 0.904·44-s − 1.17·46-s − 1.02·47-s + 0.282·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41895 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41895 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−14.79162398306541, −14.29047010526864, −13.92478195705830, −13.62196975361356, −12.99099766273929, −12.43384475260446, −11.92242416053673, −11.72143320919532, −10.98384182880248, −10.30073629810823, −9.833999493348878, −9.156637155947707, −8.720738591718327, −7.979132519611100, −7.292011231707679, −6.602686140272123, −6.200356727100317, −5.746627949035212, −5.006300677475566, −4.668778046367403, −3.739387487634896, −3.507927354984715, −2.787509376225228, −1.894110312271258, −1.333656103569807, 0,
1.333656103569807, 1.894110312271258, 2.787509376225228, 3.507927354984715, 3.739387487634896, 4.668778046367403, 5.006300677475566, 5.746627949035212, 6.200356727100317, 6.602686140272123, 7.292011231707679, 7.979132519611100, 8.720738591718327, 9.156637155947707, 9.833999493348878, 10.30073629810823, 10.98384182880248, 11.72143320919532, 11.92242416053673, 12.43384475260446, 12.99099766273929, 13.62196975361356, 13.92478195705830, 14.29047010526864, 14.79162398306541
