L(s) = 1 | − 3-s − 7-s + 9-s − 3·11-s − 2·13-s + 19-s + 21-s + 23-s − 27-s − 2·31-s + 3·33-s − 2·37-s + 2·39-s − 5·41-s + 8·43-s − 7·47-s + 49-s + 9·53-s − 57-s − 4·59-s + 2·61-s − 63-s − 4·67-s − 69-s − 8·71-s − 4·73-s + 3·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s − 0.554·13-s + 0.229·19-s + 0.218·21-s + 0.208·23-s − 0.192·27-s − 0.359·31-s + 0.522·33-s − 0.328·37-s + 0.320·39-s − 0.780·41-s + 1.21·43-s − 1.02·47-s + 1/7·49-s + 1.23·53-s − 0.132·57-s − 0.520·59-s + 0.256·61-s − 0.125·63-s − 0.488·67-s − 0.120·69-s − 0.949·71-s − 0.468·73-s + 0.341·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6882114072\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6882114072\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−13.67226411106326, −13.13836199739928, −12.91776779943497, −12.27079935325600, −11.87212721378330, −11.38505204077480, −10.77547881955018, −10.30667564468580, −10.03710158606284, −9.324377866813700, −8.926122420572910, −8.212051893051540, −7.691260812409179, −7.167524736211924, −6.791218467921957, −6.026647655575965, −5.626931143255791, −5.072295488757222, −4.599551314188767, −3.919876283097133, −3.218058443139521, −2.665289471431796, −1.990947807016523, −1.176595179005456, −0.2865800599326256,
0.2865800599326256, 1.176595179005456, 1.990947807016523, 2.665289471431796, 3.218058443139521, 3.919876283097133, 4.599551314188767, 5.072295488757222, 5.626931143255791, 6.026647655575965, 6.791218467921957, 7.167524736211924, 7.691260812409179, 8.212051893051540, 8.926122420572910, 9.324377866813700, 10.03710158606284, 10.30667564468580, 10.77547881955018, 11.38505204077480, 11.87212721378330, 12.27079935325600, 12.91776779943497, 13.13836199739928, 13.67226411106326