| L(s) = 1 | + 1.59e6·2-s + 3.59e11·4-s − 3.20e14·5-s + 2.35e17·7-s − 2.94e18·8-s − 5.13e20·10-s − 4.09e21·11-s + 2.26e22·13-s + 3.76e23·14-s − 5.49e24·16-s + 2.18e25·17-s − 1.65e26·19-s − 1.15e26·20-s − 6.54e27·22-s − 1.04e28·23-s + 5.75e28·25-s + 3.61e28·26-s + 8.45e28·28-s + 1.48e29·29-s + 2.32e30·31-s − 2.32e30·32-s + 3.48e31·34-s − 7.55e31·35-s − 5.37e31·37-s − 2.64e32·38-s + 9.44e32·40-s + 9.27e32·41-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.163·4-s − 1.50·5-s + 1.11·7-s − 0.902·8-s − 1.62·10-s − 1.83·11-s + 0.330·13-s + 1.20·14-s − 1.13·16-s + 1.30·17-s − 1.00·19-s − 0.245·20-s − 1.97·22-s − 1.27·23-s + 1.26·25-s + 0.356·26-s + 0.182·28-s + 0.155·29-s + 0.621·31-s − 0.323·32-s + 1.40·34-s − 1.67·35-s − 0.381·37-s − 1.08·38-s + 1.35·40-s + 0.804·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(42-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+41/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(21)\) |
\(\approx\) |
\(1.821327966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.821327966\) |
| \(L(\frac{43}{2})\) |
|
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 | \( 1 - 1.59e6T + 2.19e12T^{2} \) |
| 5 | \( 1 + 3.20e14T + 4.54e28T^{2} \) |
| 7 | \( 1 - 2.35e17T + 4.45e34T^{2} \) |
| 11 | \( 1 + 4.09e21T + 4.97e42T^{2} \) |
| 13 | \( 1 - 2.26e22T + 4.69e45T^{2} \) |
| 17 | \( 1 - 2.18e25T + 2.80e50T^{2} \) |
| 19 | \( 1 + 1.65e26T + 2.68e52T^{2} \) |
| 23 | \( 1 + 1.04e28T + 6.77e55T^{2} \) |
| 29 | \( 1 - 1.48e29T + 9.08e59T^{2} \) |
| 31 | \( 1 - 2.32e30T + 1.39e61T^{2} \) |
| 37 | \( 1 + 5.37e31T + 1.97e64T^{2} \) |
| 41 | \( 1 - 9.27e32T + 1.33e66T^{2} \) |
| 43 | \( 1 - 4.47e33T + 9.38e66T^{2} \) |
| 47 | \( 1 - 1.02e34T + 3.59e68T^{2} \) |
| 53 | \( 1 - 2.32e35T + 4.95e70T^{2} \) |
| 59 | \( 1 - 1.00e35T + 4.02e72T^{2} \) |
| 61 | \( 1 - 1.56e36T + 1.57e73T^{2} \) |
| 67 | \( 1 + 5.00e36T + 7.39e74T^{2} \) |
| 71 | \( 1 - 7.95e37T + 7.97e75T^{2} \) |
| 73 | \( 1 + 1.56e38T + 2.49e76T^{2} \) |
| 79 | \( 1 - 1.20e39T + 6.34e77T^{2} \) |
| 83 | \( 1 + 2.45e39T + 4.81e78T^{2} \) |
| 89 | \( 1 - 1.06e40T + 8.41e79T^{2} \) |
| 97 | \( 1 - 2.08e40T + 2.86e81T^{2} \) |
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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
−12.68799995249369474837978313136, −11.79648657061490850566163890633, −10.60077796947927563383814448638, −8.331020463592550236672127453641, −7.68797501040805993089930870565, −5.67598560151129529596397977482, −4.64159519385288998489741452472, −3.78619763391205899425437370767, −2.50869485758533505581275231364, −0.54275648870626797426047474666,
0.54275648870626797426047474666, 2.50869485758533505581275231364, 3.78619763391205899425437370767, 4.64159519385288998489741452472, 5.67598560151129529596397977482, 7.68797501040805993089930870565, 8.331020463592550236672127453641, 10.60077796947927563383814448638, 11.79648657061490850566163890633, 12.68799995249369474837978313136
