L(s) = 1 | − 2.10·2-s + 2.41·4-s − 2.10·5-s − 0.870·8-s + 4.41·10-s + 3.84·11-s + 6.82·13-s − 2.99·16-s − 1.23·17-s + 4.41·19-s − 5.07·20-s − 8.07·22-s − 6.81·23-s − 0.585·25-s − 14.3·26-s − 8.40·29-s − 0.656·31-s + 8.04·32-s + 2.58·34-s + 3.24·37-s − 9.27·38-s + 1.82·40-s + 5.07·41-s − 7.07·43-s + 9.27·44-s + 14.3·46-s + 10.6·47-s + ⋯ |
L(s) = 1 | − 1.48·2-s + 1.20·4-s − 0.939·5-s − 0.307·8-s + 1.39·10-s + 1.15·11-s + 1.89·13-s − 0.749·16-s − 0.298·17-s + 1.01·19-s − 1.13·20-s − 1.72·22-s − 1.42·23-s − 0.117·25-s − 2.81·26-s − 1.56·29-s − 0.117·31-s + 1.42·32-s + 0.443·34-s + 0.533·37-s − 1.50·38-s + 0.289·40-s + 0.792·41-s − 1.07·43-s + 1.39·44-s + 2.11·46-s + 1.55·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6922656457\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6922656457\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 5 | \( 1 + 2.10T + 5T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 - 6.82T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 - 4.41T + 19T^{2} \) |
| 23 | \( 1 + 6.81T + 23T^{2} \) |
| 29 | \( 1 + 8.40T + 29T^{2} \) |
| 31 | \( 1 + 0.656T + 31T^{2} \) |
| 37 | \( 1 - 3.24T + 37T^{2} \) |
| 41 | \( 1 - 5.07T + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 - 10.6T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 2.97T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 14.7T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 5.89T + 79T^{2} \) |
| 83 | \( 1 - 0.720T + 83T^{2} \) |
| 89 | \( 1 + 0.149T + 89T^{2} \) |
| 97 | \( 1 + 1.65T + 97T^{2} \) |
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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
−9.350972677607926639965328608393, −8.918026967518119186185273795580, −8.064502861979874884688199388609, −7.56577312547784370298594416204, −6.61710306976616621455701494829, −5.78878725162735550954208754453, −4.12855425516994439071979503820, −3.63160699092215927574436569516, −1.86063760019623894860488019417, −0.792466556965372655585308910725,
0.792466556965372655585308910725, 1.86063760019623894860488019417, 3.63160699092215927574436569516, 4.12855425516994439071979503820, 5.78878725162735550954208754453, 6.61710306976616621455701494829, 7.56577312547784370298594416204, 8.064502861979874884688199388609, 8.918026967518119186185273795580, 9.350972677607926639965328608393