L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 2·7-s + 8-s + 9-s + 10-s + 2·11-s + 12-s + 2·13-s − 2·14-s + 15-s + 16-s + 18-s + 20-s − 2·21-s + 2·22-s − 23-s + 24-s + 25-s + 2·26-s + 27-s − 2·28-s + 10·29-s + 30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.603·11-s + 0.288·12-s + 0.554·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s + 0.223·20-s − 0.436·21-s + 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.377·28-s + 1.85·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 249090 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249090 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.89449257389924, −12.80293033283778, −12.41461504223389, −11.77597421715038, −11.30318317841320, −10.76374657649937, −10.25393726358753, −9.958016658852678, −9.284757729220075, −8.854461193153371, −8.633836559693608, −7.797171476397395, −7.327138429358764, −6.896001635864793, −6.344874868298294, −5.957874803176204, −5.546513437591470, −4.784636617847892, −4.284586914727886, −3.788936871301968, −3.328540577462460, −2.723760847730519, −2.347709851030214, −1.481448510033660, −1.107856749775343, 0,
1.107856749775343, 1.481448510033660, 2.347709851030214, 2.723760847730519, 3.328540577462460, 3.788936871301968, 4.284586914727886, 4.784636617847892, 5.546513437591470, 5.957874803176204, 6.344874868298294, 6.896001635864793, 7.327138429358764, 7.797171476397395, 8.633836559693608, 8.854461193153371, 9.284757729220075, 9.958016658852678, 10.25393726358753, 10.76374657649937, 11.30318317841320, 11.77597421715038, 12.41461504223389, 12.80293033283778, 12.89449257389924