| L(s) = 1 | − 3-s − 2·4-s − 3·5-s + 4·7-s + 9-s + 3·11-s + 2·12-s − 13-s + 3·15-s + 4·16-s − 19-s + 6·20-s − 4·21-s − 9·23-s + 4·25-s − 27-s − 8·28-s − 6·29-s − 2·31-s − 3·33-s − 12·35-s − 2·36-s + 4·37-s + 39-s + 3·41-s − 7·43-s − 6·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.34·5-s + 1.51·7-s + 1/3·9-s + 0.904·11-s + 0.577·12-s − 0.277·13-s + 0.774·15-s + 16-s − 0.229·19-s + 1.34·20-s − 0.872·21-s − 1.87·23-s + 4/5·25-s − 0.192·27-s − 1.51·28-s − 1.11·29-s − 0.359·31-s − 0.522·33-s − 2.02·35-s − 1/3·36-s + 0.657·37-s + 0.160·39-s + 0.468·41-s − 1.06·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−9.721668236319765682540445072382, −8.747358907070908179484393186464, −7.960547248828429430124687340941, −7.53511853023675535658862215085, −6.12714514378066180257024905951, −5.04692212052117448633199558052, −4.31112644939450006748261353184, −3.76588854630918978975078187598, −1.57155610960796422893905590944, 0,
1.57155610960796422893905590944, 3.76588854630918978975078187598, 4.31112644939450006748261353184, 5.04692212052117448633199558052, 6.12714514378066180257024905951, 7.53511853023675535658862215085, 7.960547248828429430124687340941, 8.747358907070908179484393186464, 9.721668236319765682540445072382
