L(s) = 1 | − 14.8·2-s − 140.·3-s − 290.·4-s + 1.53e3·5-s + 2.09e3·6-s + 2.13e3·7-s + 1.19e4·8-s + 89.0·9-s − 2.28e4·10-s − 8.40e4·11-s + 4.08e4·12-s + 1.27e4·13-s − 3.18e4·14-s − 2.16e5·15-s − 2.91e4·16-s − 1.32e3·18-s + 7.81e5·19-s − 4.46e5·20-s − 3.00e5·21-s + 1.25e6·22-s − 1.32e6·23-s − 1.67e6·24-s + 4.11e5·25-s − 1.90e5·26-s + 2.75e6·27-s − 6.21e5·28-s + 2.18e6·29-s + ⋯ |
L(s) = 1 | − 0.657·2-s − 1.00·3-s − 0.567·4-s + 1.10·5-s + 0.659·6-s + 0.336·7-s + 1.03·8-s + 0.00452·9-s − 0.723·10-s − 1.73·11-s + 0.568·12-s + 0.123·13-s − 0.221·14-s − 1.10·15-s − 0.111·16-s − 0.00297·18-s + 1.37·19-s − 0.624·20-s − 0.337·21-s + 1.13·22-s − 0.987·23-s − 1.03·24-s + 0.210·25-s − 0.0815·26-s + 0.997·27-s − 0.190·28-s + 0.574·29-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
\(L(5)\) |
\(\approx\) |
\(0.6848048726\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6848048726\) |
\(L(\frac{11}{2})\) |
|
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 | \( 1 + 14.8T + 512T^{2} \) |
| 3 | \( 1 + 140.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.53e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.13e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.40e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.27e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 7.81e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.32e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.91e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.62e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.92e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 9.73e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.22e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.42e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.10e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 8.49e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.27e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.51e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.63e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.86e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.83e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.62e8T + 7.60e17T^{2} \) |
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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
−10.18906016624305858612150989174, −9.537241928418430547813886343446, −8.327388460588409270905562614857, −7.55634740051065818916732511313, −6.05450715968787787570667226656, −5.38479067680351531571737485557, −4.66122449035957000042009833659, −2.79545116984323184988714790658, −1.50644001599458201783775657657, −0.45134260017187393968568191869,
0.45134260017187393968568191869, 1.50644001599458201783775657657, 2.79545116984323184988714790658, 4.66122449035957000042009833659, 5.38479067680351531571737485557, 6.05450715968787787570667226656, 7.55634740051065818916732511313, 8.327388460588409270905562614857, 9.537241928418430547813886343446, 10.18906016624305858612150989174