| L(s) = 1 | + 4.43·2-s + 4.60·3-s + 11.7·4-s − 10.4·5-s + 20.4·6-s − 27.1·7-s + 16.4·8-s − 5.78·9-s − 46.2·10-s − 51.5·11-s + 53.9·12-s − 7.02·13-s − 120.·14-s − 47.9·15-s − 20.6·16-s + 106.·17-s − 25.6·18-s + 75.7·19-s − 121.·20-s − 124.·21-s − 228.·22-s + 75.7·24-s − 16.4·25-s − 31.1·26-s − 151.·27-s − 317.·28-s − 123.·29-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.886·3-s + 1.46·4-s − 0.931·5-s + 1.39·6-s − 1.46·7-s + 0.726·8-s − 0.214·9-s − 1.46·10-s − 1.41·11-s + 1.29·12-s − 0.149·13-s − 2.29·14-s − 0.825·15-s − 0.322·16-s + 1.52·17-s − 0.336·18-s + 0.915·19-s − 1.36·20-s − 1.29·21-s − 2.21·22-s + 0.644·24-s − 0.131·25-s − 0.235·26-s − 1.07·27-s − 2.14·28-s − 0.790·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 \) |
| good | 2 | \( 1 - 4.43T + 8T^{2} \) |
| 3 | \( 1 - 4.60T + 27T^{2} \) |
| 5 | \( 1 + 10.4T + 125T^{2} \) |
| 7 | \( 1 + 27.1T + 343T^{2} \) |
| 11 | \( 1 + 51.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.02T + 2.19e3T^{2} \) |
| 17 | \( 1 - 106.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 75.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 54.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 77.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 286.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 171.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 450.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 89.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 580.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 108.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 51.8T + 3.00e5T^{2} \) |
| 71 | \( 1 + 437.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 764.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 568.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 378.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 181.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 578.T + 9.12e5T^{2} \) |
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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
−10.02724587714869312391515519010, −9.139803661749481371976516931352, −7.82097531255883583185866347526, −7.34766268473602565570344708403, −5.95456370207506578473387325787, −5.27380403151348333019670140040, −3.83292766543494752416889010867, −3.26437883371539953952135536249, −2.62441605352675501530091767207, 0,
2.62441605352675501530091767207, 3.26437883371539953952135536249, 3.83292766543494752416889010867, 5.27380403151348333019670140040, 5.95456370207506578473387325787, 7.34766268473602565570344708403, 7.82097531255883583185866347526, 9.139803661749481371976516931352, 10.02724587714869312391515519010
