| L(s) = 1 | − 3.95·2-s + 182.·3-s − 496.·4-s + 659.·5-s − 722.·6-s + 1.39e3·7-s + 3.98e3·8-s + 1.36e4·9-s − 2.61e3·10-s + 1.98e4·11-s − 9.06e4·12-s + 1.37e5·13-s − 5.50e3·14-s + 1.20e5·15-s + 2.38e5·16-s − 5.40e4·18-s + 5.46e5·19-s − 3.27e5·20-s + 2.54e5·21-s − 7.86e4·22-s + 1.06e5·23-s + 7.28e5·24-s − 1.51e6·25-s − 5.45e5·26-s − 1.09e6·27-s − 6.90e5·28-s + 4.27e6·29-s + ⋯ |
| L(s) = 1 | − 0.174·2-s + 1.30·3-s − 0.969·4-s + 0.472·5-s − 0.227·6-s + 0.219·7-s + 0.344·8-s + 0.694·9-s − 0.0825·10-s + 0.409·11-s − 1.26·12-s + 1.33·13-s − 0.0382·14-s + 0.614·15-s + 0.909·16-s − 0.121·18-s + 0.962·19-s − 0.457·20-s + 0.285·21-s − 0.0715·22-s + 0.0796·23-s + 0.448·24-s − 0.777·25-s − 0.234·26-s − 0.397·27-s − 0.212·28-s + 1.12·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.577241721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.577241721\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 3.95T + 512T^{2} \) |
| 3 | \( 1 - 182.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 659.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.39e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 1.98e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.37e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 5.46e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.06e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.27e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.17e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 3.86e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.38e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.16e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.85e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.76e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.01e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 9.08e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.28e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.69e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.17e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 5.30e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.09e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 9.09e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.28e8T + 7.60e17T^{2} \) |
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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
−9.764494057827449474295689175802, −9.261132339038292927653739771497, −8.393136153303066029086959813619, −7.85138850781835525345018256511, −6.35499888256387747466164657146, −5.16087251862757458827923619851, −3.93253275446043528595150994673, −3.20014789485123932252843943154, −1.82191492954301902265608501208, −0.866979095872531128163227156689,
0.866979095872531128163227156689, 1.82191492954301902265608501208, 3.20014789485123932252843943154, 3.93253275446043528595150994673, 5.16087251862757458827923619851, 6.35499888256387747466164657146, 7.85138850781835525345018256511, 8.393136153303066029086959813619, 9.261132339038292927653739771497, 9.764494057827449474295689175802
