L(s) = 1 | + (−2.54 − 4.40i)2-s + (3.08 − 5.33i)4-s + (20.9 + 36.2i)5-s + (−127. + 25.2i)7-s − 193.·8-s + (106. − 184. i)10-s + (36.0 − 62.4i)11-s − 632.·13-s + (434. + 495. i)14-s + (394. + 683. i)16-s + (−987. + 1.71e3i)17-s + (932. + 1.61e3i)19-s + 257.·20-s − 366.·22-s + (206. + 358. i)23-s + ⋯ |
L(s) = 1 | + (−0.449 − 0.778i)2-s + (0.0963 − 0.166i)4-s + (0.374 + 0.647i)5-s + (−0.980 + 0.194i)7-s − 1.07·8-s + (0.336 − 0.582i)10-s + (0.0898 − 0.155i)11-s − 1.03·13-s + (0.592 + 0.675i)14-s + (0.385 + 0.667i)16-s + (−0.829 + 1.43i)17-s + (0.592 + 1.02i)19-s + 0.144·20-s − 0.161·22-s + (0.0815 + 0.141i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.132 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.231780 + 0.264765i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.231780 + 0.264765i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (127. - 25.2i)T \) |
good | 2 | \( 1 + (2.54 + 4.40i)T + (-16 + 27.7i)T^{2} \) |
| 5 | \( 1 + (-20.9 - 36.2i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-36.0 + 62.4i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + 632.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (987. - 1.71e3i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-932. - 1.61e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-206. - 358. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + 731.T + 2.05e7T^{2} \) |
| 31 | \( 1 + (3.06e3 - 5.30e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (5.17e3 + 8.96e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 - 3.52e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.45e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.07e4 + 1.85e4i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-6.28e3 + 1.08e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.80e4 - 3.12e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (2.01e3 + 3.48e3i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (7.78e3 - 1.34e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.21e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (9.79e3 - 1.69e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.80e4 + 3.12e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 - 2.45e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (3.51e4 + 6.08e4i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.05e5T + 8.58e9T^{2} \) |
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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
−14.36978427380195443985950356024, −12.86646995168249289393100286301, −11.92844222642365824583954889327, −10.54776121950410922423217440207, −10.00006840313375839032572918113, −8.818165706125496889832270657627, −6.86895219639836137764741346849, −5.80111432001345467078370952337, −3.34937414094138546247709352559, −1.98629366673045850786259681431,
0.17387037174013408892591958288, 2.85446907406613355330191360649, 5.00947575547540185369129840272, 6.59963245529876785773131606849, 7.47352532906445551643652992223, 9.098347115431822718935336626382, 9.612213936171578514783608211178, 11.53409189820901215731998301775, 12.66925079140809371636724775686, 13.60313141736047300224875638822