| L(s) = 1 | − 4·2-s + 3·3-s + 8·4-s + 5·5-s − 12·6-s + 9·9-s − 20·10-s − 10·11-s + 24·12-s + 24·13-s + 15·15-s − 64·16-s − 54·17-s − 36·18-s + 12·19-s + 40·20-s + 40·22-s − 134·23-s + 25·25-s − 96·26-s + 27·27-s + 118·29-s − 60·30-s − 144·31-s + 256·32-s − 30·33-s + 216·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 1/3·9-s − 0.632·10-s − 0.274·11-s + 0.577·12-s + 0.512·13-s + 0.258·15-s − 16-s − 0.770·17-s − 0.471·18-s + 0.144·19-s + 0.447·20-s + 0.387·22-s − 1.21·23-s + 1/5·25-s − 0.724·26-s + 0.192·27-s + 0.755·29-s − 0.365·30-s − 0.834·31-s + 1.41·32-s − 0.158·33-s + 1.08·34-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 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 11 | \( 1 + 10 T + p^{3} T^{2} \) |
| 13 | \( 1 - 24 T + p^{3} T^{2} \) |
| 17 | \( 1 + 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 12 T + p^{3} T^{2} \) |
| 23 | \( 1 + 134 T + p^{3} T^{2} \) |
| 29 | \( 1 - 118 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 + 378 T + p^{3} T^{2} \) |
| 41 | \( 1 + 330 T + p^{3} T^{2} \) |
| 43 | \( 1 - 204 T + p^{3} T^{2} \) |
| 47 | \( 1 + 312 T + p^{3} T^{2} \) |
| 53 | \( 1 - 142 T + p^{3} T^{2} \) |
| 59 | \( 1 + 108 T + p^{3} T^{2} \) |
| 61 | \( 1 - 168 T + p^{3} T^{2} \) |
| 67 | \( 1 - 448 T + p^{3} T^{2} \) |
| 71 | \( 1 - 146 T + p^{3} T^{2} \) |
| 73 | \( 1 - 360 T + p^{3} T^{2} \) |
| 79 | \( 1 - 236 T + p^{3} T^{2} \) |
| 83 | \( 1 + 324 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1728 T + p^{3} T^{2} \) |
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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
−9.480659797804585593718927978598, −8.670900788983564009427560468805, −8.194644673621459862176508738044, −7.19222856787250364733165032688, −6.39488130968039116377572278853, −5.05835272605674629255752644468, −3.76863189502786890152355488643, −2.35317554155400038608283770181, −1.47739609947507904212208552367, 0,
1.47739609947507904212208552367, 2.35317554155400038608283770181, 3.76863189502786890152355488643, 5.05835272605674629255752644468, 6.39488130968039116377572278853, 7.19222856787250364733165032688, 8.194644673621459862176508738044, 8.670900788983564009427560468805, 9.480659797804585593718927978598
