| L(s) = 1 | + 1.73·2-s − 5·4-s − 12.1·5-s − 22·7-s − 22.5·8-s − 21·10-s + 58.8·11-s − 49·13-s − 38.1·14-s + 1.00·16-s + 36.3·17-s − 70·19-s + 60.6·20-s + 101.·22-s + 24.2·23-s + 22·25-s − 84.8·26-s + 110·28-s − 278.·29-s − 112·31-s + 181.·32-s + 63·34-s + 266.·35-s + 281·37-s − 121.·38-s + 273·40-s − 48.4·41-s + ⋯ |
| L(s) = 1 | + 0.612·2-s − 0.625·4-s − 1.08·5-s − 1.18·7-s − 0.995·8-s − 0.664·10-s + 1.61·11-s − 1.04·13-s − 0.727·14-s + 0.0156·16-s + 0.518·17-s − 0.845·19-s + 0.677·20-s + 0.988·22-s + 0.219·23-s + 0.175·25-s − 0.640·26-s + 0.742·28-s − 1.78·29-s − 0.648·31-s + 1.00·32-s + 0.317·34-s + 1.28·35-s + 1.24·37-s − 0.517·38-s + 1.07·40-s − 0.184·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 8T^{2} \) |
| 5 | \( 1 + 12.1T + 125T^{2} \) |
| 7 | \( 1 + 22T + 343T^{2} \) |
| 11 | \( 1 - 58.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49T + 2.19e3T^{2} \) |
| 17 | \( 1 - 36.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 70T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 112T + 2.97e4T^{2} \) |
| 37 | \( 1 - 281T + 5.06e4T^{2} \) |
| 41 | \( 1 + 48.4T + 6.89e4T^{2} \) |
| 43 | \( 1 - 50T + 7.95e4T^{2} \) |
| 47 | \( 1 + 242.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 374.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 96.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 679T + 2.26e5T^{2} \) |
| 67 | \( 1 + 274T + 3.00e5T^{2} \) |
| 71 | \( 1 + 446.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 511T + 3.89e5T^{2} \) |
| 79 | \( 1 + 526T + 4.93e5T^{2} \) |
| 83 | \( 1 + 387.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 36.3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.77e3T + 9.12e5T^{2} \) |
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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
−13.15995712429744555116308612514, −12.38042181225703724143146957743, −11.55149289936280562329640042957, −9.758684047212521677458891953231, −8.942896381163664759099998333096, −7.36239258677455043181188058848, −6.04482244989724912855888469316, −4.31398066036974669701650558322, −3.43507782157423252473590756521, 0,
3.43507782157423252473590756521, 4.31398066036974669701650558322, 6.04482244989724912855888469316, 7.36239258677455043181188058848, 8.942896381163664759099998333096, 9.758684047212521677458891953231, 11.55149289936280562329640042957, 12.38042181225703724143146957743, 13.15995712429744555116308612514
