L(s) = 1 | + 2-s + 1.61·3-s + 4-s − 5-s + 1.61·6-s − 7-s + 8-s − 0.381·9-s − 10-s + 1.61·12-s + 5.23·13-s − 14-s − 1.61·15-s + 16-s − 2.61·17-s − 0.381·18-s − 7.85·19-s − 20-s − 1.61·21-s − 0.472·23-s + 1.61·24-s + 25-s + 5.23·26-s − 5.47·27-s − 28-s − 4.47·29-s − 1.61·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.934·3-s + 0.5·4-s − 0.447·5-s + 0.660·6-s − 0.377·7-s + 0.353·8-s − 0.127·9-s − 0.316·10-s + 0.467·12-s + 1.45·13-s − 0.267·14-s − 0.417·15-s + 0.250·16-s − 0.634·17-s − 0.0900·18-s − 1.80·19-s − 0.223·20-s − 0.353·21-s − 0.0984·23-s + 0.330·24-s + 0.200·25-s + 1.02·26-s − 1.05·27-s − 0.188·28-s − 0.830·29-s − 0.295·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 + 2.61T + 17T^{2} \) |
| 19 | \( 1 + 7.85T + 19T^{2} \) |
| 23 | \( 1 + 0.472T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 - 2.09T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 + 2.76T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 3.38T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 2.85T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 8.85T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 8.38T + 97T^{2} \) |
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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
−7.45034175310107062002090660040, −6.61527408571093802816398623923, −6.17344506689355132454695042873, −5.35998975045016349058030706891, −4.31757136168024228817471570209, −3.82549103373577736432633238536, −3.24936618577266879197907249320, −2.40087215492088515732067346828, −1.62478535356923383953808221363, 0,
1.62478535356923383953808221363, 2.40087215492088515732067346828, 3.24936618577266879197907249320, 3.82549103373577736432633238536, 4.31757136168024228817471570209, 5.35998975045016349058030706891, 6.17344506689355132454695042873, 6.61527408571093802816398623923, 7.45034175310107062002090660040