L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 11-s + 6·13-s + 16-s + 4·19-s + 20-s + 22-s + 25-s − 6·26-s + 6·29-s − 32-s + 2·37-s − 4·38-s − 40-s − 6·41-s − 4·43-s − 44-s + 8·47-s − 7·49-s − 50-s + 6·52-s − 14·53-s − 55-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 0.301·11-s + 1.66·13-s + 1/4·16-s + 0.917·19-s + 0.223·20-s + 0.213·22-s + 1/5·25-s − 1.17·26-s + 1.11·29-s − 0.176·32-s + 0.328·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s − 0.150·44-s + 1.16·47-s − 49-s − 0.141·50-s + 0.832·52-s − 1.92·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.98802473032274, −12.37290178810672, −12.07394375522816, −11.35894335465788, −11.08323071991725, −10.68834554983276, −10.12894576961018, −9.771484935264986, −9.218721312995885, −8.877025363529398, −8.278580133688963, −7.980249790188521, −7.515300735771638, −6.732124425662105, −6.422036398712253, −6.085447785306785, −5.362412317947688, −5.018443094945242, −4.302803169856568, −3.601129622850711, −3.153077038091697, −2.681695423676577, −1.861061320697125, −1.360743843533922, −0.9078702215464194, 0,
0.9078702215464194, 1.360743843533922, 1.861061320697125, 2.681695423676577, 3.153077038091697, 3.601129622850711, 4.302803169856568, 5.018443094945242, 5.362412317947688, 6.085447785306785, 6.422036398712253, 6.732124425662105, 7.515300735771638, 7.980249790188521, 8.278580133688963, 8.877025363529398, 9.218721312995885, 9.771484935264986, 10.12894576961018, 10.68834554983276, 11.08323071991725, 11.35894335465788, 12.07394375522816, 12.37290178810672, 12.98802473032274