L(s) = 1 | + (0.264 − 0.457i)2-s + (2.94 − 0.546i)3-s + (1.86 + 3.22i)4-s + (−4.68 − 1.75i)5-s + (0.529 − 1.49i)6-s + (−2.39 − 1.38i)7-s + 4.08·8-s + (8.40 − 3.22i)9-s + (−2.04 + 1.67i)10-s + (−7.99 − 4.61i)11-s + (7.24 + 8.48i)12-s + (−11.7 + 6.79i)13-s + (−1.26 + 0.731i)14-s + (−14.7 − 2.62i)15-s + (−6.36 + 11.0i)16-s − 12.2·17-s + ⋯ |
L(s) = 1 | + (0.132 − 0.228i)2-s + (0.983 − 0.182i)3-s + (0.465 + 0.805i)4-s + (−0.936 − 0.351i)5-s + (0.0883 − 0.249i)6-s + (−0.342 − 0.197i)7-s + 0.510·8-s + (0.933 − 0.357i)9-s + (−0.204 + 0.167i)10-s + (−0.726 − 0.419i)11-s + (0.603 + 0.707i)12-s + (−0.905 + 0.522i)13-s + (−0.0904 + 0.0522i)14-s + (−0.984 − 0.174i)15-s + (−0.397 + 0.688i)16-s − 0.718·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.40601 - 0.117880i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40601 - 0.117880i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.94 + 0.546i)T \) |
| 5 | \( 1 + (4.68 + 1.75i)T \) |
good | 2 | \( 1 + (-0.264 + 0.457i)T + (-2 - 3.46i)T^{2} \) |
| 7 | \( 1 + (2.39 + 1.38i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (7.99 + 4.61i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (11.7 - 6.79i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 12.2T + 289T^{2} \) |
| 19 | \( 1 - 20.2T + 361T^{2} \) |
| 23 | \( 1 + (1.18 + 2.05i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-30.2 - 17.4i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (14.7 + 25.5i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 64.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-34.5 + 19.9i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-58.5 - 33.7i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (46.6 - 80.8i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 9.82T + 2.80e3T^{2} \) |
| 59 | \( 1 + (50.6 - 29.2i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-7.75 + 13.4i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (13.4 - 7.78i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 53.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 23.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (17.2 - 29.9i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-37.6 + 65.2i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 29.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (54.0 + 31.1i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−15.82406133417776109674801060416, −14.34988023059611628533205272807, −13.06212560849777992483157702863, −12.31302701591581993569723275414, −11.02461538845961976685157999713, −9.244305641542999919329328819313, −7.937803375248305517496227805558, −7.16788844944430378453908824579, −4.24942103024816315101217291651, −2.82838803753688120300136203740,
2.79513498110861090978731268244, 4.83830000470124971150424687242, 6.94136758034735891291139845978, 7.956068606337192378920493339616, 9.672050815107388559028241195618, 10.65530705427724062589397478723, 12.18296345313442327297921198168, 13.63148361637056166578240002235, 14.79554694496912695655794488696, 15.47380617012099135701414491814