| L(s) = 1 | + 1.59·2-s − 5.44·4-s + 18.2·5-s − 21.4·8-s + 29.2·10-s − 61.2·11-s − 32.4·13-s + 9.23·16-s − 81.3·17-s + 20.9·19-s − 99.6·20-s − 97.9·22-s + 33.7·23-s + 209.·25-s − 51.8·26-s − 52.0·29-s − 193.·31-s + 186.·32-s − 129.·34-s − 267.·37-s + 33.4·38-s − 393.·40-s + 203.·41-s − 21.9·43-s + 333.·44-s + 53.9·46-s − 247.·47-s + ⋯ |
| L(s) = 1 | + 0.564·2-s − 0.680·4-s + 1.63·5-s − 0.949·8-s + 0.924·10-s − 1.67·11-s − 0.692·13-s + 0.144·16-s − 1.16·17-s + 0.252·19-s − 1.11·20-s − 0.948·22-s + 0.305·23-s + 1.67·25-s − 0.391·26-s − 0.333·29-s − 1.12·31-s + 1.03·32-s − 0.655·34-s − 1.18·37-s + 0.142·38-s − 1.55·40-s + 0.773·41-s − 0.0778·43-s + 1.14·44-s + 0.172·46-s − 0.769·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 1.59T + 8T^{2} \) |
| 5 | \( 1 - 18.2T + 125T^{2} \) |
| 11 | \( 1 + 61.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 81.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 20.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 33.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 52.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 267.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 203.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 21.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 247.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 140.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 221.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 652.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 604.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 716.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 388.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 289.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 115.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 939.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 120.T + 9.12e5T^{2} \) |
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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
−10.16210734012911807435906689289, −9.421333633673480431541865228590, −8.676016187434296881510762399316, −7.37375341585540121241272294002, −6.11489608697863692898591598325, −5.33014805024167671628966362962, −4.72734250238702833409624480412, −3.03659359450312523268893677263, −2.03441783252030420224266247137, 0,
2.03441783252030420224266247137, 3.03659359450312523268893677263, 4.72734250238702833409624480412, 5.33014805024167671628966362962, 6.11489608697863692898591598325, 7.37375341585540121241272294002, 8.676016187434296881510762399316, 9.421333633673480431541865228590, 10.16210734012911807435906689289
