L(s) = 1 | − 2-s + 2.81·3-s + 4-s + 1.22·5-s − 2.81·6-s + 1.31·7-s − 8-s + 4.90·9-s − 1.22·10-s + 6.06·11-s + 2.81·12-s + 4.95·13-s − 1.31·14-s + 3.45·15-s + 16-s + 5.02·17-s − 4.90·18-s − 5.59·19-s + 1.22·20-s + 3.70·21-s − 6.06·22-s + 0.915·23-s − 2.81·24-s − 3.48·25-s − 4.95·26-s + 5.35·27-s + 1.31·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.62·3-s + 0.5·4-s + 0.549·5-s − 1.14·6-s + 0.498·7-s − 0.353·8-s + 1.63·9-s − 0.388·10-s + 1.82·11-s + 0.811·12-s + 1.37·13-s − 0.352·14-s + 0.892·15-s + 0.250·16-s + 1.21·17-s − 1.15·18-s − 1.28·19-s + 0.274·20-s + 0.809·21-s − 1.29·22-s + 0.190·23-s − 0.573·24-s − 0.697·25-s − 0.971·26-s + 1.03·27-s + 0.249·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.675637285\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.675637285\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2011 | \( 1 - T \) |
good | 3 | \( 1 - 2.81T + 3T^{2} \) |
| 5 | \( 1 - 1.22T + 5T^{2} \) |
| 7 | \( 1 - 1.31T + 7T^{2} \) |
| 11 | \( 1 - 6.06T + 11T^{2} \) |
| 13 | \( 1 - 4.95T + 13T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 19 | \( 1 + 5.59T + 19T^{2} \) |
| 23 | \( 1 - 0.915T + 23T^{2} \) |
| 29 | \( 1 - 3.89T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 1.90T + 37T^{2} \) |
| 41 | \( 1 - 1.88T + 41T^{2} \) |
| 43 | \( 1 - 7.34T + 43T^{2} \) |
| 47 | \( 1 - 7.69T + 47T^{2} \) |
| 53 | \( 1 + 7.06T + 53T^{2} \) |
| 59 | \( 1 + 8.37T + 59T^{2} \) |
| 61 | \( 1 - 0.249T + 61T^{2} \) |
| 67 | \( 1 + 1.69T + 67T^{2} \) |
| 71 | \( 1 + 9.42T + 71T^{2} \) |
| 73 | \( 1 + 11.1T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 + 14.2T + 83T^{2} \) |
| 89 | \( 1 - 16.8T + 89T^{2} \) |
| 97 | \( 1 - 4.04T + 97T^{2} \) |
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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
−8.708663093103304476993676547478, −7.899957744980720778887771091090, −7.28624047784055532775449657433, −6.35390311467525766727990009162, −5.78298230907266812257542811311, −4.22193271917692327508350432303, −3.73704781939405818803562637866, −2.85115476261955436749287202231, −1.64831027078157074817798665512, −1.43915989524975392325621573009,
1.43915989524975392325621573009, 1.64831027078157074817798665512, 2.85115476261955436749287202231, 3.73704781939405818803562637866, 4.22193271917692327508350432303, 5.78298230907266812257542811311, 6.35390311467525766727990009162, 7.28624047784055532775449657433, 7.899957744980720778887771091090, 8.708663093103304476993676547478