| L(s) = 1 | + (−0.635 − 1.89i)2-s + (−3.19 + 2.40i)4-s − 10.1·7-s + (6.59 + 4.52i)8-s − 10.6i·11-s − 11.7i·13-s + (6.41 + 19.1i)14-s + (4.38 − 15.3i)16-s + 16.6i·17-s − 0.464i·19-s + (−20.2 + 6.77i)22-s − 42.1·23-s + (−22.3 + 7.47i)26-s + (32.2 − 24.3i)28-s + 19.5·29-s + ⋯ |
| L(s) = 1 | + (−0.317 − 0.948i)2-s + (−0.798 + 0.602i)4-s − 1.44·7-s + (0.824 + 0.565i)8-s − 0.968i·11-s − 0.905i·13-s + (0.458 + 1.36i)14-s + (0.274 − 0.961i)16-s + 0.978i·17-s − 0.0244i·19-s + (−0.918 + 0.307i)22-s − 1.83·23-s + (−0.858 + 0.287i)26-s + (1.15 − 0.869i)28-s + 0.674·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6882488495\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6882488495\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.635 + 1.89i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 10.1T + 49T^{2} \) |
| 11 | \( 1 + 10.6iT - 121T^{2} \) |
| 13 | \( 1 + 11.7iT - 169T^{2} \) |
| 17 | \( 1 - 16.6iT - 289T^{2} \) |
| 19 | \( 1 + 0.464iT - 361T^{2} \) |
| 23 | \( 1 + 42.1T + 529T^{2} \) |
| 29 | \( 1 - 19.5T + 841T^{2} \) |
| 31 | \( 1 - 9.17iT - 961T^{2} \) |
| 37 | \( 1 - 23.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 44.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 58.3T + 1.84e3T^{2} \) |
| 47 | \( 1 + 41.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 80.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 63.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 80.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 22.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 61.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 137. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 138. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 86.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 127.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 15iT - 9.40e3T^{2} \) |
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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
−10.14742216660394231292169815846, −9.261851956660420808265928052120, −8.419867240612609778247764006827, −7.71415307206247322246134474048, −6.33824702238084299997929238304, −5.67189239523503393467944832782, −4.14994106113835497725313267668, −3.37965615783056759636555227317, −2.52650470405417631589050736495, −0.869042988132869558747113814649,
0.33866798581349021665931786643, 2.18699661084398113925376578883, 3.76681195627714124995514860581, 4.62516498460922737595519739007, 5.80983033845348048793307321975, 6.58564334659780473340798834911, 7.17281785036559639553757688544, 8.093064098444991320026752973055, 9.242420274656900737759798416345, 9.660250214196964859627901463552
