L(s) = 1 | − 1.83·2-s + 1.36·4-s + 2.83·5-s + 4.39·7-s + 1.16·8-s − 5.19·10-s + 4.39·11-s + 4·13-s − 8.06·14-s − 4.86·16-s − 4.39·17-s + 1.80·19-s + 3.86·20-s − 8.06·22-s − 2.72·23-s + 3.03·25-s − 7.33·26-s + 6.00·28-s − 2.03·29-s − 5.66·31-s + 6.59·32-s + 8.06·34-s + 12.4·35-s + 1.07·37-s − 3.30·38-s + 3.30·40-s − 8.39·41-s + ⋯ |
L(s) = 1 | − 1.29·2-s + 0.682·4-s + 1.26·5-s + 1.66·7-s + 0.412·8-s − 1.64·10-s + 1.32·11-s + 1.10·13-s − 2.15·14-s − 1.21·16-s − 1.06·17-s + 0.413·19-s + 0.864·20-s − 1.71·22-s − 0.569·23-s + 0.606·25-s − 1.43·26-s + 1.13·28-s − 0.377·29-s − 1.01·31-s + 1.16·32-s + 1.38·34-s + 2.10·35-s + 0.176·37-s − 0.535·38-s + 0.522·40-s − 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.219910358\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.219910358\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 5 | \( 1 - 2.83T + 5T^{2} \) |
| 7 | \( 1 - 4.39T + 7T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 4.39T + 17T^{2} \) |
| 19 | \( 1 - 1.80T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 + 2.03T + 29T^{2} \) |
| 31 | \( 1 + 5.66T + 31T^{2} \) |
| 37 | \( 1 - 1.07T + 37T^{2} \) |
| 41 | \( 1 + 8.39T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 3.27T + 47T^{2} \) |
| 53 | \( 1 - 3.66T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 + 8.46T + 61T^{2} \) |
| 67 | \( 1 - 3.33T + 67T^{2} \) |
| 73 | \( 1 - 2.83T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 + 5.80T + 83T^{2} \) |
| 89 | \( 1 + 0.509T + 89T^{2} \) |
| 97 | \( 1 + 6.06T + 97T^{2} \) |
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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
−10.43697254857881630745239595279, −9.553938211418492096843875298875, −8.805241375298873718440681079460, −8.368391899972415677199729153241, −7.20545193145059908982615883085, −6.27958880685359009427331582448, −5.18231934014042534505704264006, −4.06374642088540901594800944882, −1.88240119062699117415124488488, −1.45015407625119885672538038962,
1.45015407625119885672538038962, 1.88240119062699117415124488488, 4.06374642088540901594800944882, 5.18231934014042534505704264006, 6.27958880685359009427331582448, 7.20545193145059908982615883085, 8.368391899972415677199729153241, 8.805241375298873718440681079460, 9.553938211418492096843875298875, 10.43697254857881630745239595279