| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.444 - 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.444070736\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.444070736\) |
| \(L(2)\) |
|
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
−9.848179703565833627346400363960, −8.733242284411161004371147964372, −8.516980527861596855386274790591, −7.35344358951050349368868576639, −6.80021913316527358202628107373, −5.53930228314161010687703940879, −4.95129084899285574183396448150, −4.05163821730510738208887182734, −2.74413319532372420082842054345, −0.976746775108162426948036732867,
1.06412161300301148061938158646, 2.05035351051343831780784094601, 3.22756120987093223916747917404, 4.52861165763784099858501079498, 4.96462728541444110577252287011, 6.01795208012567755171831729980, 7.38382564696334725577374665826, 8.258570454762645862597156178193, 8.979845397710084104147735678572, 10.05279754005222387721651427568
