L(s) = 1 | + (1.37 − 2.37i)2-s + (12.2 + 21.1i)4-s + (29.1 − 50.5i)5-s + (−21.4 + 127. i)7-s + 155.·8-s + (−80.2 − 138. i)10-s + (8.71 + 15.0i)11-s + 889.·13-s + (274. + 226. i)14-s + (−178. + 308. i)16-s + (513. + 889. i)17-s + (869. − 1.50e3i)19-s + 1.42e3·20-s + 47.8·22-s + (1.96e3 − 3.40e3i)23-s + ⋯ |
L(s) = 1 | + (0.242 − 0.420i)2-s + (0.381 + 0.661i)4-s + (0.522 − 0.904i)5-s + (−0.165 + 0.986i)7-s + 0.856·8-s + (−0.253 − 0.439i)10-s + (0.0217 + 0.0376i)11-s + 1.46·13-s + (0.374 + 0.309i)14-s + (−0.173 + 0.301i)16-s + (0.430 + 0.746i)17-s + (0.552 − 0.957i)19-s + 0.797·20-s + 0.0210·22-s + (0.775 − 1.34i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.994 + 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.47001 - 0.126996i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.47001 - 0.126996i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (21.4 - 127. i)T \) |
good | 2 | \( 1 + (-1.37 + 2.37i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-29.1 + 50.5i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-8.71 - 15.0i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 889.T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-513. - 889. i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-869. + 1.50e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-1.96e3 + 3.40e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 5.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-1.54e3 - 2.68e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (2.51e3 - 4.35e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.83e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (4.80e3 - 8.31e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (1.16e4 + 2.01e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.80e3 + 3.12e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.14e4 - 1.98e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.35e4 + 4.07e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 1.59e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + (2.96e3 + 5.13e3i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-4.42e4 + 7.66e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 9.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (2.32e4 - 4.02e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 7.59e4T + 8.58e9T^{2} \) |
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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
−13.44745199180996289892886574073, −12.85690485658925676281368137808, −11.88228823810789006660822442664, −10.76557484746689654554130852349, −9.105021111815970588157067465406, −8.302501559144770194127863087550, −6.46680667916689134201252454535, −5.02018787831344439120887918505, −3.25916490878785512029035979922, −1.60100842539679883761423261546,
1.38095645184881750988500101384, 3.51283288971617070275976505547, 5.51930212590954880588010106326, 6.60440072420327010918554362523, 7.58263478147098886454878801947, 9.644740063336668464470147278693, 10.57791415580590568498951403150, 11.39999928392880380861640886849, 13.48963636943716245794381606601, 13.91921655509214965507532250650