| L(s) = 1 | − 3.23·2-s + 2.47·4-s − 13.5·5-s − 1.42·7-s + 17.8·8-s + 43.9·10-s + 54.5·11-s + 4.62·14-s − 77.6·16-s + 114.·17-s + 104.·19-s − 33.6·20-s − 176.·22-s + 64.5·23-s + 59.4·25-s − 3.53·28-s + 60.8·29-s + 148.·31-s + 108.·32-s − 370.·34-s + 19.4·35-s + 20.9·37-s − 339.·38-s − 242.·40-s + 371.·41-s + 40.2·43-s + 135.·44-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.309·4-s − 1.21·5-s − 0.0771·7-s + 0.790·8-s + 1.39·10-s + 1.49·11-s + 0.0883·14-s − 1.21·16-s + 1.63·17-s + 1.26·19-s − 0.375·20-s − 1.71·22-s + 0.585·23-s + 0.475·25-s − 0.0238·28-s + 0.389·29-s + 0.862·31-s + 0.598·32-s − 1.86·34-s + 0.0937·35-s + 0.0930·37-s − 1.44·38-s − 0.960·40-s + 1.41·41-s + 0.142·43-s + 0.462·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.091464820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.091464820\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.23T + 8T^{2} \) |
| 5 | \( 1 + 13.5T + 125T^{2} \) |
| 7 | \( 1 + 1.42T + 343T^{2} \) |
| 11 | \( 1 - 54.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 114.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 64.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 60.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 148.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 20.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 371.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 40.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 639.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 102.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 704.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 819.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 574.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 365.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 965.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 580.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 175.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 20.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.22e3T + 9.12e5T^{2} \) |
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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
−9.315249447043483362395798277906, −8.226896426911406126205882792725, −7.68039711310275315335735405666, −7.12735326520648250326639348445, −6.02924267619786665672729368708, −4.76907211898739670745312561934, −3.94622937877904622400151499542, −3.08431338008795127515806839819, −1.29410533260458139784285910312, −0.73125201241811418804149476542,
0.73125201241811418804149476542, 1.29410533260458139784285910312, 3.08431338008795127515806839819, 3.94622937877904622400151499542, 4.76907211898739670745312561934, 6.02924267619786665672729368708, 7.12735326520648250326639348445, 7.68039711310275315335735405666, 8.226896426911406126205882792725, 9.315249447043483362395798277906
