L(s) = 1 | − 2-s + 3-s + 4-s + 2.64·5-s − 6-s − 7-s − 8-s + 9-s − 2.64·10-s + 0.671·11-s + 12-s − 13-s + 14-s + 2.64·15-s + 16-s + 17-s − 18-s − 5.18·19-s + 2.64·20-s − 21-s − 0.671·22-s − 7.73·23-s − 24-s + 1.97·25-s + 26-s + 27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.18·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.835·10-s + 0.202·11-s + 0.288·12-s − 0.277·13-s + 0.267·14-s + 0.682·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s − 1.18·19-s + 0.590·20-s − 0.218·21-s − 0.143·22-s − 1.61·23-s − 0.204·24-s + 0.395·25-s + 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 - 2.64T + 5T^{2} \) |
| 11 | \( 1 - 0.671T + 11T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 + 7.73T + 23T^{2} \) |
| 29 | \( 1 + 9.10T + 29T^{2} \) |
| 31 | \( 1 - 8.06T + 31T^{2} \) |
| 37 | \( 1 + 6.52T + 37T^{2} \) |
| 41 | \( 1 - 3.56T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 - 4.02T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 5.64T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 + 6.76T + 67T^{2} \) |
| 71 | \( 1 + 8.34T + 71T^{2} \) |
| 73 | \( 1 - 6.93T + 73T^{2} \) |
| 79 | \( 1 - 0.655T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−7.42922837710753552801612960556, −6.78330926701169355584186011008, −5.94793367311658529409833539718, −5.73297125512678071365664273892, −4.45155944780094366635943452936, −3.75338845336260191702663974663, −2.69165230575145439162393291466, −2.13481225498254063871951234657, −1.43814020157290143842405709084, 0,
1.43814020157290143842405709084, 2.13481225498254063871951234657, 2.69165230575145439162393291466, 3.75338845336260191702663974663, 4.45155944780094366635943452936, 5.73297125512678071365664273892, 5.94793367311658529409833539718, 6.78330926701169355584186011008, 7.42922837710753552801612960556