| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 6.57·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s + 13.1·10-s + 18.0·11-s − 12·12-s + 13·13-s − 14·14-s − 19.7·15-s + 16·16-s − 42.3·17-s + 18·18-s + 77.7·19-s + 26.2·20-s + 21·21-s + 36.1·22-s + 43.8·23-s − 24·24-s − 81.7·25-s + 26·26-s − 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.587·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.415·10-s + 0.495·11-s − 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.339·15-s + 0.250·16-s − 0.604·17-s + 0.235·18-s + 0.938·19-s + 0.293·20-s + 0.218·21-s + 0.350·22-s + 0.397·23-s − 0.204·24-s − 0.654·25-s + 0.196·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.947111048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.947111048\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 6.57T + 125T^{2} \) |
| 11 | \( 1 - 18.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 42.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 77.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 43.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 121.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 248.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 425.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 179.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 216.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 713.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 403.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 246.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 742.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 65.6T + 3.57e5T^{2} \) |
| 73 | \( 1 - 965.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 382.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 830.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 376.T + 9.12e5T^{2} \) |
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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
−10.45463581685724531050055255133, −9.727606991301275538461913356755, −8.707232962088113423144034869413, −7.35749669280534342649152082388, −6.48403379467893319070742225867, −5.81237069493611635216232959761, −4.83034962277324929651688602657, −3.75886214893734267646206258598, −2.46267946334579441020744160968, −1.01953215186912515275215253221,
1.01953215186912515275215253221, 2.46267946334579441020744160968, 3.75886214893734267646206258598, 4.83034962277324929651688602657, 5.81237069493611635216232959761, 6.48403379467893319070742225867, 7.35749669280534342649152082388, 8.707232962088113423144034869413, 9.727606991301275538461913356755, 10.45463581685724531050055255133
