L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 1.08·7-s − 8-s + 9-s − 10-s + 1.41·11-s − 12-s − 3.84·13-s + 1.08·14-s − 15-s + 16-s − 18-s + 2.82·19-s + 20-s + 1.08·21-s − 1.41·22-s − 2.98·23-s + 24-s + 25-s + 3.84·26-s − 27-s − 1.08·28-s − 8.64·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.447·5-s + 0.408·6-s − 0.409·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s + 0.426·11-s − 0.288·12-s − 1.06·13-s + 0.289·14-s − 0.258·15-s + 0.250·16-s − 0.235·18-s + 0.648·19-s + 0.223·20-s + 0.236·21-s − 0.301·22-s − 0.621·23-s + 0.204·24-s + 0.200·25-s + 0.754·26-s − 0.192·27-s − 0.204·28-s − 1.60·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 1.08T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 3.84T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 2.98T + 23T^{2} \) |
| 29 | \( 1 + 8.64T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 - 7.39T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 + 4.27T + 43T^{2} \) |
| 47 | \( 1 - 9.39T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 2.46T + 61T^{2} \) |
| 67 | \( 1 - 4.95T + 67T^{2} \) |
| 71 | \( 1 - 3.24T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 + 0.328T + 79T^{2} \) |
| 83 | \( 1 + 0.537T + 83T^{2} \) |
| 89 | \( 1 - 17.5T + 89T^{2} \) |
| 97 | \( 1 - 0.253T + 97T^{2} \) |
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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
−7.58467081274768842512869910520, −6.71933684734667008824030115006, −6.13095748048526713659355702388, −5.56010012637990803890095705594, −4.72185970736607588839806672374, −3.86798742941644350314266838798, −2.85477906505455176729078697623, −2.07382779778469706258643527600, −1.08724554637480176459523834249, 0,
1.08724554637480176459523834249, 2.07382779778469706258643527600, 2.85477906505455176729078697623, 3.86798742941644350314266838798, 4.72185970736607588839806672374, 5.56010012637990803890095705594, 6.13095748048526713659355702388, 6.71933684734667008824030115006, 7.58467081274768842512869910520