| L(s) = 1 | + 3·2-s + 4-s − 20·7-s − 21·8-s + 24·11-s − 74·13-s − 60·14-s − 71·16-s + 54·17-s − 124·19-s + 72·22-s − 120·23-s − 222·26-s − 20·28-s + 78·29-s + 200·31-s − 45·32-s + 162·34-s + 70·37-s − 372·38-s − 330·41-s − 92·43-s + 24·44-s − 360·46-s − 24·47-s + 57·49-s − 74·52-s + ⋯ |
| L(s) = 1 | + 1.06·2-s + 1/8·4-s − 1.07·7-s − 0.928·8-s + 0.657·11-s − 1.57·13-s − 1.14·14-s − 1.10·16-s + 0.770·17-s − 1.49·19-s + 0.697·22-s − 1.08·23-s − 1.67·26-s − 0.134·28-s + 0.499·29-s + 1.15·31-s − 0.248·32-s + 0.817·34-s + 0.311·37-s − 1.58·38-s − 1.25·41-s − 0.326·43-s + 0.0822·44-s − 1.15·46-s − 0.0744·47-s + 0.166·49-s − 0.197·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 3 T + p^{3} T^{2} \) |
| 7 | \( 1 + 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 200 T + p^{3} T^{2} \) |
| 37 | \( 1 - 70 T + p^{3} T^{2} \) |
| 41 | \( 1 + 330 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 - 450 T + p^{3} T^{2} \) |
| 59 | \( 1 + 24 T + p^{3} T^{2} \) |
| 61 | \( 1 + 322 T + p^{3} T^{2} \) |
| 67 | \( 1 - 196 T + p^{3} T^{2} \) |
| 71 | \( 1 - 288 T + p^{3} T^{2} \) |
| 73 | \( 1 - 430 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 - 286 T + p^{3} T^{2} \) |
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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
−11.90778819164220862100965240576, −10.19144311086461557443487565667, −9.559974877637067924580253906881, −8.348596345906738696540401957759, −6.82730598190250880689978538633, −6.06613246028294637687707169883, −4.80750741579903131949499602018, −3.78677782881581625327031002249, −2.57978758470335437435358267536, 0,
2.57978758470335437435358267536, 3.78677782881581625327031002249, 4.80750741579903131949499602018, 6.06613246028294637687707169883, 6.82730598190250880689978538633, 8.348596345906738696540401957759, 9.559974877637067924580253906881, 10.19144311086461557443487565667, 11.90778819164220862100965240576
