| L(s) = 1 | − 1.27e3·2-s + 5.90e4·3-s − 4.79e5·4-s − 2.12e7·5-s − 7.51e7·6-s + 6.32e8·7-s + 3.27e9·8-s + 3.48e9·9-s + 2.70e10·10-s + 5.97e10·11-s − 2.83e10·12-s + 7.38e11·13-s − 8.03e11·14-s − 1.25e12·15-s − 3.16e12·16-s − 8.35e12·17-s − 4.43e12·18-s + 4.19e13·19-s + 1.01e13·20-s + 3.73e13·21-s − 7.60e13·22-s + 4.48e13·23-s + 1.93e14·24-s − 2.41e13·25-s − 9.39e14·26-s + 2.05e14·27-s − 3.02e14·28-s + ⋯ |
| L(s) = 1 | − 0.878·2-s + 0.577·3-s − 0.228·4-s − 0.974·5-s − 0.507·6-s + 0.845·7-s + 1.07·8-s + 0.333·9-s + 0.855·10-s + 0.694·11-s − 0.131·12-s + 1.48·13-s − 0.742·14-s − 0.562·15-s − 0.719·16-s − 1.00·17-s − 0.292·18-s + 1.56·19-s + 0.222·20-s + 0.488·21-s − 0.610·22-s + 0.225·23-s + 0.622·24-s − 0.0505·25-s − 1.30·26-s + 0.192·27-s − 0.193·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.158187133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.158187133\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 5.90e4T \) |
| 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 | \( 1 + 5.92e20T + 5.27e41T^{2} \) |
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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
−20.48939555480983151977093474340, −19.15820348472221603750848389109, −17.80162857818988804969123027532, −15.78466180859828723067671639827, −13.83399071979712557391409054935, −11.24907079689894967943496641418, −8.985468658839268506463878719954, −7.74641459196408745592655202908, −4.09558009622312387136400257069, −1.12394556401439308955174821673,
1.12394556401439308955174821673, 4.09558009622312387136400257069, 7.74641459196408745592655202908, 8.985468658839268506463878719954, 11.24907079689894967943496641418, 13.83399071979712557391409054935, 15.78466180859828723067671639827, 17.80162857818988804969123027532, 19.15820348472221603750848389109, 20.48939555480983151977093474340
