L(s) = 1 | − 0.566·2-s − 0.484·3-s − 1.67·4-s + 5-s + 0.274·6-s − 0.0113·7-s + 2.08·8-s − 2.76·9-s − 0.566·10-s − 0.725·11-s + 0.814·12-s − 3.41·13-s + 0.00644·14-s − 0.484·15-s + 2.17·16-s + 3.34·17-s + 1.56·18-s + 2.32·19-s − 1.67·20-s + 0.00552·21-s + 0.410·22-s − 4.69·23-s − 1.01·24-s + 25-s + 1.93·26-s + 2.79·27-s + 0.0191·28-s + ⋯ |
L(s) = 1 | − 0.400·2-s − 0.279·3-s − 0.839·4-s + 0.447·5-s + 0.112·6-s − 0.00430·7-s + 0.736·8-s − 0.921·9-s − 0.179·10-s − 0.218·11-s + 0.235·12-s − 0.947·13-s + 0.00172·14-s − 0.125·15-s + 0.544·16-s + 0.811·17-s + 0.369·18-s + 0.534·19-s − 0.375·20-s + 0.00120·21-s + 0.0876·22-s − 0.978·23-s − 0.206·24-s + 0.200·25-s + 0.379·26-s + 0.537·27-s + 0.00361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8399872932\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8399872932\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 + 0.566T + 2T^{2} \) |
| 3 | \( 1 + 0.484T + 3T^{2} \) |
| 7 | \( 1 + 0.0113T + 7T^{2} \) |
| 11 | \( 1 + 0.725T + 11T^{2} \) |
| 13 | \( 1 + 3.41T + 13T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 - 2.32T + 19T^{2} \) |
| 23 | \( 1 + 4.69T + 23T^{2} \) |
| 29 | \( 1 - 5.86T + 29T^{2} \) |
| 31 | \( 1 + 0.631T + 31T^{2} \) |
| 37 | \( 1 - 3.86T + 37T^{2} \) |
| 41 | \( 1 + 0.320T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 0.842T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 2.52T + 61T^{2} \) |
| 67 | \( 1 - 8.96T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 5.27T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 + 0.282T + 83T^{2} \) |
| 89 | \( 1 - 4.26T + 89T^{2} \) |
| 97 | \( 1 - 17.9T + 97T^{2} \) |
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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
−9.869079024380288329733399691379, −8.971115897830264653073070371965, −8.182877874842671368561188259347, −7.51651279263336421581108799506, −6.29437131375763023651746465579, −5.40152078600835956778283322753, −4.83433982563804232294390933937, −3.56423659310736034310406884521, −2.36087948535182102354076580035, −0.73396573744702646470768207377,
0.73396573744702646470768207377, 2.36087948535182102354076580035, 3.56423659310736034310406884521, 4.83433982563804232294390933937, 5.40152078600835956778283322753, 6.29437131375763023651746465579, 7.51651279263336421581108799506, 8.182877874842671368561188259347, 8.971115897830264653073070371965, 9.869079024380288329733399691379