| L(s) = 1 | + 2-s − 2.11·3-s + 4-s + 2.85·5-s − 2.11·6-s + 0.696·7-s + 8-s + 1.49·9-s + 2.85·10-s − 11-s − 2.11·12-s − 3.28·13-s + 0.696·14-s − 6.06·15-s + 16-s − 1.84·17-s + 1.49·18-s + 3.98·19-s + 2.85·20-s − 1.47·21-s − 22-s − 3.01·23-s − 2.11·24-s + 3.17·25-s − 3.28·26-s + 3.19·27-s + 0.696·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.22·3-s + 0.5·4-s + 1.27·5-s − 0.865·6-s + 0.263·7-s + 0.353·8-s + 0.497·9-s + 0.904·10-s − 0.301·11-s − 0.611·12-s − 0.910·13-s + 0.186·14-s − 1.56·15-s + 0.250·16-s − 0.447·17-s + 0.351·18-s + 0.915·19-s + 0.639·20-s − 0.322·21-s − 0.213·22-s − 0.629·23-s − 0.432·24-s + 0.635·25-s − 0.643·26-s + 0.615·27-s + 0.131·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.427458015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.427458015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 + T \) |
| good | 3 | \( 1 + 2.11T + 3T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 - 0.696T + 7T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 + 1.84T + 17T^{2} \) |
| 19 | \( 1 - 3.98T + 19T^{2} \) |
| 23 | \( 1 + 3.01T + 23T^{2} \) |
| 29 | \( 1 - 0.0188T + 29T^{2} \) |
| 31 | \( 1 - 2.31T + 31T^{2} \) |
| 37 | \( 1 - 3.68T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 + 0.815T + 47T^{2} \) |
| 53 | \( 1 - 1.95T + 53T^{2} \) |
| 59 | \( 1 - 14.7T + 59T^{2} \) |
| 61 | \( 1 - 7.02T + 61T^{2} \) |
| 67 | \( 1 + 6.04T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 - 8.67T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 - 5.90T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−8.227330675442459959425252570070, −7.36107256197069455320029952806, −6.60270252990162298144610205855, −5.98298892443705470889577137093, −5.41043569056204579840387850708, −4.97377398551013860791921224948, −4.10011197308122406003662125173, −2.74104530341006537908233644667, −2.08867458995112205570602938680, −0.840327744796629212035139659373,
0.840327744796629212035139659373, 2.08867458995112205570602938680, 2.74104530341006537908233644667, 4.10011197308122406003662125173, 4.97377398551013860791921224948, 5.41043569056204579840387850708, 5.98298892443705470889577137093, 6.60270252990162298144610205855, 7.36107256197069455320029952806, 8.227330675442459959425252570070
