| L(s) = 1 | + (−0.591 + 2.15i)5-s + 2.69i·7-s − 0.968i·11-s − 2.99i·13-s − 1.02·17-s + 7.89i·19-s − 0.0832i·23-s + (−4.29 − 2.55i)25-s + (−4.28 − 3.25i)29-s + 2.02i·31-s + (−5.81 − 1.59i)35-s + 3.06·37-s + 3.92i·41-s − 7.15·43-s + 0.554·47-s + ⋯ |
| L(s) = 1 | + (−0.264 + 0.964i)5-s + 1.01i·7-s − 0.291i·11-s − 0.831i·13-s − 0.249·17-s + 1.81i·19-s − 0.0173i·23-s + (−0.859 − 0.510i)25-s + (−0.796 − 0.604i)29-s + 0.363i·31-s + (−0.983 − 0.269i)35-s + 0.503·37-s + 0.613i·41-s − 1.09·43-s + 0.0808·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3359883719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3359883719\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.591 - 2.15i)T \) |
| 29 | \( 1 + (4.28 + 3.25i)T \) |
| good | 7 | \( 1 - 2.69iT - 7T^{2} \) |
| 11 | \( 1 + 0.968iT - 11T^{2} \) |
| 13 | \( 1 + 2.99iT - 13T^{2} \) |
| 17 | \( 1 + 1.02T + 17T^{2} \) |
| 19 | \( 1 - 7.89iT - 19T^{2} \) |
| 23 | \( 1 + 0.0832iT - 23T^{2} \) |
| 31 | \( 1 - 2.02iT - 31T^{2} \) |
| 37 | \( 1 - 3.06T + 37T^{2} \) |
| 41 | \( 1 - 3.92iT - 41T^{2} \) |
| 43 | \( 1 + 7.15T + 43T^{2} \) |
| 47 | \( 1 - 0.554T + 47T^{2} \) |
| 53 | \( 1 + 6.25iT - 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 - 8.77iT - 61T^{2} \) |
| 67 | \( 1 + 2.40iT - 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 - 6.55iT - 79T^{2} \) |
| 83 | \( 1 + 7.10iT - 83T^{2} \) |
| 89 | \( 1 + 6.83iT - 89T^{2} \) |
| 97 | \( 1 - 4.18T + 97T^{2} \) |
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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
−8.440866519984943571054662768387, −7.963318498552394825206605854138, −7.31213683032015144815176885430, −6.25294137800130840337624848175, −5.94704282473884271667856972421, −5.15934334347246533837046599688, −4.01703583208055843863404906977, −3.29322570919226375473968722690, −2.59522326269529533449130679488, −1.63683266822024441043464242315,
0.092051216766765740728830424629, 1.14364805732313749406575629008, 2.12989143859381292141297498115, 3.34857548113760011289539615750, 4.28194473839164463349335149042, 4.62430559079127763781271749176, 5.43309345957566896229503914113, 6.50335600873075520083524070104, 7.13694393638585073384234710401, 7.67178159834842219630330756967
