| L(s) = 1 | + 3·2-s + 3·3-s + 4-s + 5·5-s + 9·6-s − 21·8-s + 9·9-s + 15·10-s − 45·11-s + 3·12-s + 31·13-s + 15·15-s − 71·16-s − 96·17-s + 27·18-s − 149·19-s + 5·20-s − 135·22-s − 141·23-s − 63·24-s + 25·25-s + 93·26-s + 27·27-s + 48·29-s + 45·30-s + 178·31-s − 45·32-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.447·5-s + 0.612·6-s − 0.928·8-s + 1/3·9-s + 0.474·10-s − 1.23·11-s + 0.0721·12-s + 0.661·13-s + 0.258·15-s − 1.10·16-s − 1.36·17-s + 0.353·18-s − 1.79·19-s + 0.0559·20-s − 1.30·22-s − 1.27·23-s − 0.535·24-s + 1/5·25-s + 0.701·26-s + 0.192·27-s + 0.307·29-s + 0.273·30-s + 1.03·31-s − 0.248·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 11 | \( 1 + 45 T + p^{3} T^{2} \) |
| 13 | \( 1 - 31 T + p^{3} T^{2} \) |
| 17 | \( 1 + 96 T + p^{3} T^{2} \) |
| 19 | \( 1 + 149 T + p^{3} T^{2} \) |
| 23 | \( 1 + 141 T + p^{3} T^{2} \) |
| 29 | \( 1 - 48 T + p^{3} T^{2} \) |
| 31 | \( 1 - 178 T + p^{3} T^{2} \) |
| 37 | \( 1 - 371 T + p^{3} T^{2} \) |
| 41 | \( 1 + 225 T + p^{3} T^{2} \) |
| 43 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 375 T + p^{3} T^{2} \) |
| 53 | \( 1 + 663 T + p^{3} T^{2} \) |
| 59 | \( 1 - 60 T + p^{3} T^{2} \) |
| 61 | \( 1 + 392 T + p^{3} T^{2} \) |
| 67 | \( 1 + 280 T + p^{3} T^{2} \) |
| 71 | \( 1 - 258 T + p^{3} T^{2} \) |
| 73 | \( 1 + 578 T + p^{3} T^{2} \) |
| 79 | \( 1 - 152 T + p^{3} T^{2} \) |
| 83 | \( 1 - 432 T + p^{3} T^{2} \) |
| 89 | \( 1 - 234 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1352 T + p^{3} T^{2} \) |
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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
−9.533177363541209949975476988325, −8.592782123313783476553513670388, −8.017662774947834714071640372511, −6.48373185214944751352592394510, −6.02874135821687978532105499998, −4.72481794560135905803331751494, −4.18722411899679317479048433406, −2.89422595207186455987804387232, −2.10560218382305686602904372427, 0,
2.10560218382305686602904372427, 2.89422595207186455987804387232, 4.18722411899679317479048433406, 4.72481794560135905803331751494, 6.02874135821687978532105499998, 6.48373185214944751352592394510, 8.017662774947834714071640372511, 8.592782123313783476553513670388, 9.533177363541209949975476988325
