| L(s) = 1 | + 2-s − 2.74·3-s + 4-s − 2.60·5-s − 2.74·6-s + 0.636·7-s + 8-s + 4.53·9-s − 2.60·10-s + 2.71·11-s − 2.74·12-s − 4.45·13-s + 0.636·14-s + 7.14·15-s + 16-s + 6.79·17-s + 4.53·18-s + 5.23·19-s − 2.60·20-s − 1.74·21-s + 2.71·22-s − 7.82·23-s − 2.74·24-s + 1.76·25-s − 4.45·26-s − 4.22·27-s + 0.636·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.58·3-s + 0.5·4-s − 1.16·5-s − 1.12·6-s + 0.240·7-s + 0.353·8-s + 1.51·9-s − 0.822·10-s + 0.818·11-s − 0.792·12-s − 1.23·13-s + 0.170·14-s + 1.84·15-s + 0.250·16-s + 1.64·17-s + 1.06·18-s + 1.19·19-s − 0.581·20-s − 0.381·21-s + 0.578·22-s − 1.63·23-s − 0.560·24-s + 0.352·25-s − 0.873·26-s − 0.812·27-s + 0.120·28-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
| \(L(1)\) |
\(\approx\) |
\(1.140107589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.140107589\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.74T + 3T^{2} \) |
| 5 | \( 1 + 2.60T + 5T^{2} \) |
| 7 | \( 1 - 0.636T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 + 4.45T + 13T^{2} \) |
| 17 | \( 1 - 6.79T + 17T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 + 7.82T + 23T^{2} \) |
| 29 | \( 1 + 9.41T + 29T^{2} \) |
| 31 | \( 1 + 5.38T + 31T^{2} \) |
| 37 | \( 1 - 3.98T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 - 3.47T + 53T^{2} \) |
| 59 | \( 1 + 2.87T + 59T^{2} \) |
| 61 | \( 1 + 9.83T + 61T^{2} \) |
| 67 | \( 1 - 6.31T + 67T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 + 2.28T + 79T^{2} \) |
| 83 | \( 1 - 8.33T + 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 - 5.01T + 97T^{2} \) |
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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
−7.85733882401337223271878314771, −7.65414555850664133173981861842, −6.92569833305768628138876942472, −5.96552130090250724579419585023, −5.47005252049676020818943177488, −4.79706487702072633424152158418, −3.97097468884644959537980636702, −3.39568719312990612287267362995, −1.81101627216485637119135374433, −0.59932499904075790482764595645,
0.59932499904075790482764595645, 1.81101627216485637119135374433, 3.39568719312990612287267362995, 3.97097468884644959537980636702, 4.79706487702072633424152158418, 5.47005252049676020818943177488, 5.96552130090250724579419585023, 6.92569833305768628138876942472, 7.65414555850664133173981861842, 7.85733882401337223271878314771
