| L(s) = 1 | + 1.27e3·2-s − 4.79e5·4-s + 2.12e7·5-s + 6.32e8·7-s − 3.27e9·8-s + 2.70e10·10-s − 5.97e10·11-s + 7.38e11·13-s + 8.03e11·14-s − 3.16e12·16-s + 8.35e12·17-s + 4.19e13·19-s − 1.01e13·20-s − 7.60e13·22-s − 4.48e13·23-s − 2.41e13·25-s + 9.39e14·26-s − 3.02e14·28-s + 2.76e15·29-s + 8.36e15·31-s + 2.84e15·32-s + 1.06e16·34-s + 1.34e16·35-s − 1.77e16·37-s + 5.33e16·38-s − 6.97e16·40-s − 1.45e17·41-s + ⋯ |
| L(s) = 1 | + 0.878·2-s − 0.228·4-s + 0.974·5-s + 0.845·7-s − 1.07·8-s + 0.855·10-s − 0.694·11-s + 1.48·13-s + 0.742·14-s − 0.719·16-s + 1.00·17-s + 1.56·19-s − 0.222·20-s − 0.610·22-s − 0.225·23-s − 0.0505·25-s + 1.30·26-s − 0.193·28-s + 1.22·29-s + 1.83·31-s + 0.447·32-s + 0.883·34-s + 0.824·35-s − 0.607·37-s + 1.37·38-s − 1.05·40-s − 1.69·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(3.564208383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.564208383\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 1.27e3T + 2.09e6T^{2} \) |
| 5 | \( 1 - 2.12e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 6.32e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 5.97e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 7.38e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 8.35e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 4.19e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 4.48e13T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.76e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 8.36e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 1.77e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.45e17T + 7.38e33T^{2} \) |
| 43 | \( 1 - 1.24e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 4.28e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 4.77e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 1.61e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 3.76e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.81e18T + 2.22e38T^{2} \) |
| 71 | \( 1 + 1.00e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 1.72e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 3.28e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 3.05e17T + 1.99e40T^{2} \) |
| 89 | \( 1 + 2.34e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 5.92e20T + 5.27e41T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.71950741249817679745562138072, −14.07258330141691945639913971775, −13.52145812722002435240906711502, −11.85078522177591995827296343640, −10.01631058338208857695815856530, −8.327728500524252218697218219586, −5.98136708677993779512077036755, −4.93956968454570441006330021795, −3.14767765435172049491292905417, −1.24415797490543756439724998456,
1.24415797490543756439724998456, 3.14767765435172049491292905417, 4.93956968454570441006330021795, 5.98136708677993779512077036755, 8.327728500524252218697218219586, 10.01631058338208857695815856530, 11.85078522177591995827296343640, 13.52145812722002435240906711502, 14.07258330141691945639913971775, 15.71950741249817679745562138072
