| L(s) = 1 | − 3.91·2-s − 5.59·3-s + 11.3·4-s + 21.9·6-s + 7·7-s − 28.6·8-s + 22.3·9-s − 63.3·12-s − 10.6·13-s − 27.4·14-s + 66.9·16-s − 8.62·17-s − 87.3·18-s − 15.6·19-s − 39.1·21-s − 8.23·23-s + 160.·24-s + 25·25-s + 41.5·26-s − 74.4·27-s + 79.2·28-s − 147.·32-s + 33.7·34-s + 252.·36-s + 73.2·37-s + 61.2·38-s + 59.3·39-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 1.86·3-s + 2.83·4-s + 3.65·6-s + 7-s − 3.58·8-s + 2.47·9-s − 5.27·12-s − 0.815·13-s − 1.95·14-s + 4.18·16-s − 0.507·17-s − 4.85·18-s − 0.823·19-s − 1.86·21-s − 0.358·23-s + 6.68·24-s + 25-s + 1.59·26-s − 2.75·27-s + 2.83·28-s − 4.60·32-s + 0.993·34-s + 7.01·36-s + 1.97·37-s + 1.61·38-s + 1.52·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3132582388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3132582388\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 + 3.91T + 4T^{2} \) |
| 3 | \( 1 + 5.59T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 + 10.6T + 169T^{2} \) |
| 17 | \( 1 + 8.62T + 289T^{2} \) |
| 19 | \( 1 + 15.6T + 361T^{2} \) |
| 23 | \( 1 + 8.23T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 73.2T + 1.36e3T^{2} \) |
| 43 | \( 1 + 81.4T + 1.84e3T^{2} \) |
| 47 | \( 1 - 83.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 42.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 159.T + 9.40e3T^{2} \) |
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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
−11.36352694608000281388806069325, −10.57900797427342075497194776102, −10.02066613110187588019382922402, −8.803167959641116131950551495064, −7.68428145308919603804676311021, −6.86590931150006811970774793054, −6.00720526215114104098249464801, −4.76861295400151735513184941129, −1.99634797256106063676769407557, −0.66865295643309513387791194345,
0.66865295643309513387791194345, 1.99634797256106063676769407557, 4.76861295400151735513184941129, 6.00720526215114104098249464801, 6.86590931150006811970774793054, 7.68428145308919603804676311021, 8.803167959641116131950551495064, 10.02066613110187588019382922402, 10.57900797427342075497194776102, 11.36352694608000281388806069325
