L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.766 − 0.642i)11-s + (0.173 + 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.766 + 0.642i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.939 − 0.342i)20-s + (0.766 − 0.642i)22-s + (0.939 + 0.342i)23-s + ⋯ |
L(s) = 1 | + (−0.173 + 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)5-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (−0.5 − 0.866i)10-s + (−0.766 − 0.642i)11-s + (0.173 + 0.984i)13-s + (−0.173 − 0.984i)14-s + (0.766 + 0.642i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.939 − 0.342i)20-s + (0.766 − 0.642i)22-s + (0.939 + 0.342i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.05921041265 + 0.5065774553i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.05921041265 + 0.5065774553i\) |
\(L(1)\) |
\(\approx\) |
\(0.4465541098 + 0.4213020739i\) |
\(L(1)\) |
\(\approx\) |
\(0.4465541098 + 0.4213020739i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 7 | \( 1 + (-0.939 + 0.342i)T \) |
| 11 | \( 1 + (-0.766 - 0.642i)T \) |
| 13 | \( 1 + (0.173 + 0.984i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.173 + 0.984i)T \) |
| 31 | \( 1 + (-0.939 - 0.342i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.939 - 0.342i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (-0.766 + 0.642i)T \) |
| 61 | \( 1 + (-0.939 + 0.342i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (-0.173 + 0.984i)T \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.766 + 0.642i)T \) |
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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
−36.71054444770414453315070241699, −35.89273433976354570983526116374, −34.7838087009068832282993933988, −32.584456174519239319836971505542, −31.696014406351162532665383735159, −30.45846737693771441945897715639, −29.10439627877956792165226807424, −28.09090131467346742700868665622, −26.94782156542666580240618103479, −25.535973708211900024866295605071, −23.44910100705191497046355503754, −22.60941689075382846974945049000, −20.77015092437506355351238214470, −19.92909377559328978111419403560, −18.69728937412319122803231942658, −17.06790930075876897197040109459, −15.534029608088021273197870062108, −13.26198261813935985602005333439, −12.40138710650043564277517233115, −10.74735226845145709356964508007, −9.30272849829234547207330912408, −7.67624332891158225658961573183, −4.83919863193021852520397902, −3.11869405203256950744374240630, −0.40515979689655816385167494069,
3.67587321190866865665264571594, 5.912213949868429947069378661919, 7.31391374102176597511236372497, 8.8489952580935580482103306558, 10.606114965047337592231145829, 12.71382006953854536147982547109, 14.37634047361713257153511885309, 15.65784272347344243117203224705, 16.64921086860621634899223254886, 18.64107535239525937157780119572, 19.19716449836050570360196533730, 21.68004100222232808790321897485, 23.027631564054719980470720924423, 23.89912739879143438241082614699, 25.63015970793499628409406678245, 26.42362686748262217914851674450, 27.68663959252193311963148303947, 29.161769214902919219806521372071, 31.120543154667379999215267918855, 31.89060273007651327608339253228, 33.37914373523119456545712606541, 34.6032551363486151966910720753, 35.324007812268473141980116406429, 36.65414954387836433063715957172, 38.14748585470791634191199758469