| L(s) = 1 | + (−4.89 − 2.82i)2-s + (15.9 + 27.7i)4-s + (29.3 − 16.9i)5-s + (102.5 − 177. i)7-s − 181. i·8-s − 192·10-s + (911. + 526. i)11-s + (1.02e3 + 1.76e3i)13-s + (−1.00e3 + 579. i)14-s + (−512. + 886. i)16-s − 8.24e3i·17-s − 1.50e3·19-s + (940. + 543. i)20-s + (−2.97e3 − 5.15e3i)22-s + (−6.14e3 + 3.54e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.235 − 0.135i)5-s + (0.298 − 0.517i)7-s − 0.353i·8-s − 0.192·10-s + (0.684 + 0.395i)11-s + (0.464 + 0.804i)13-s + (−0.365 + 0.211i)14-s + (−0.125 + 0.216i)16-s − 1.67i·17-s − 0.218·19-s + (0.117 + 0.0678i)20-s + (−0.279 − 0.484i)22-s + (−0.504 + 0.291i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.679044743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.679044743\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 + 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-29.3 + 16.9i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-102.5 + 177. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-911. - 526. i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.02e3 - 1.76e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 8.24e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 1.50e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (6.14e3 - 3.54e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.46e4 - 1.42e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-1.74e4 - 3.03e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 5.76e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-1.16e5 + 6.71e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.12e4 + 5.41e4i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (4.49e4 + 2.59e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + 7.71e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-3.21e5 + 1.85e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-3.06e4 + 5.30e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (3.38e4 + 5.86e4i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + 5.09e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 4.23e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (-3.53e5 + 6.12e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-7.45e5 - 4.30e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + 6.67e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (2.63e5 - 4.55e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−11.58299460278176947664334055804, −10.60881590501826212645966826106, −9.521730631847216706715546160392, −8.820249780447624812859310103229, −7.46692523733391224029657346008, −6.61796700151419464634557335735, −4.93069333726196727619013610646, −3.64287154508504065228150373495, −2.00748743883235904696039615780, −0.817596573994817419314402719903,
0.948238583198703305343084625756, 2.36233648232392568331749008188, 4.08830520441828417752229253542, 5.81610949614541924458306271498, 6.37951479169710624735764366172, 8.051332853582129432717594961206, 8.541821748832189284790528811491, 9.837207274393317924958139330615, 10.67759247702765114239902104650, 11.72068221630444428005862001579
