L(s) = 1 | + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (0.781 + 0.623i)11-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.623 − 0.781i)16-s + i·17-s + (0.433 + 0.900i)19-s + (−0.623 − 0.781i)20-s + (0.900 + 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯ |
L(s) = 1 | + (0.974 − 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.781 − 0.623i)8-s + (−0.433 − 0.900i)10-s + (0.781 + 0.623i)11-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)14-s + (0.623 − 0.781i)16-s + i·17-s + (0.433 + 0.900i)19-s + (−0.623 − 0.781i)20-s + (0.900 + 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.421755791 - 0.6513343394i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.421755791 - 0.6513343394i\) |
\(L(1)\) |
\(\approx\) |
\(1.515852990 - 0.4500496064i\) |
\(L(1)\) |
\(\approx\) |
\(1.515852990 - 0.4500496064i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.974 - 0.222i)T \) |
| 5 | \( 1 + (-0.222 - 0.974i)T \) |
| 7 | \( 1 + (-0.900 - 0.433i)T \) |
| 11 | \( 1 + (0.781 + 0.623i)T \) |
| 13 | \( 1 + (-0.623 + 0.781i)T \) |
| 17 | \( 1 + iT \) |
| 19 | \( 1 + (0.433 + 0.900i)T \) |
| 23 | \( 1 + (0.222 - 0.974i)T \) |
| 31 | \( 1 + (-0.974 + 0.222i)T \) |
| 37 | \( 1 + (-0.781 + 0.623i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + (0.974 + 0.222i)T \) |
| 47 | \( 1 + (-0.781 - 0.623i)T \) |
| 53 | \( 1 + (0.222 + 0.974i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (-0.433 + 0.900i)T \) |
| 67 | \( 1 + (-0.623 - 0.781i)T \) |
| 71 | \( 1 + (0.623 - 0.781i)T \) |
| 73 | \( 1 + (-0.974 - 0.222i)T \) |
| 79 | \( 1 + (0.781 - 0.623i)T \) |
| 83 | \( 1 + (0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.974 - 0.222i)T \) |
| 97 | \( 1 + (-0.433 - 0.900i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−30.89026172398750874494775550922, −29.742793875657619518929915012241, −29.25903499174205135062456765449, −27.519076565622611064780442134231, −26.31082661598456412128344109357, −25.35119678428309843541619662355, −24.431029211545119037423290038712, −23.03243733098151225684881066217, −22.32488662400290816585896809259, −21.67675365073721216477205350048, −19.99771609141464441824491696895, −19.19911887378523928454212084481, −17.68864723266802555300610930589, −16.218509719605149305172125816746, −15.3569082334792576304779636973, −14.31837270470587908452909627881, −13.23859370399916758783451173633, −11.953040253261293724822240752341, −11.012147489921689583595193797597, −9.44966914895003161058044106723, −7.52102882701628915212348136024, −6.574743851215106038782826367061, −5.38185497151867450100838436564, −3.54527762116227231774438990762, −2.71411951507047086895996879193,
1.680058947455952566698098077860, 3.66492229145882049030833676956, 4.608553182447082972319010892505, 6.12615541320523178926089779469, 7.33442938340726552147533383001, 9.19707778446097620071014503596, 10.4067920096107605338908308694, 12.09772707907527712811988299711, 12.577531439107180311650838311023, 13.84644044675218120637247718965, 14.9944223897229628148512182522, 16.34269154690480482507510727691, 16.94900454072427332517522125785, 19.16279486027204184152134600077, 19.91965201615822405811827190441, 20.82540970352698969528802332031, 22.08530589564730710724929133546, 22.99824430664483791647103225897, 24.033397634976995751827616436969, 24.894037138004055213298481323429, 26.04063018964394281476730281887, 27.61958631095917095647895629853, 28.74398701160786533537529161645, 29.363587569558026691042331963509, 30.71240359934784811776354312433