L(s) = 1 | + (−0.733 − 0.680i)5-s + (0.826 + 0.563i)11-s + (−0.0747 + 0.997i)13-s + (0.623 + 0.781i)17-s + 19-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.365 + 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.623 + 0.781i)37-s + (−0.733 − 0.680i)41-s + (0.733 − 0.680i)43-s + (−0.826 − 0.563i)47-s + (−0.623 + 0.781i)53-s + (−0.222 − 0.974i)55-s + ⋯ |
L(s) = 1 | + (−0.733 − 0.680i)5-s + (0.826 + 0.563i)11-s + (−0.0747 + 0.997i)13-s + (0.623 + 0.781i)17-s + 19-s + (0.365 + 0.930i)23-s + (0.0747 + 0.997i)25-s + (−0.365 + 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.623 + 0.781i)37-s + (−0.733 − 0.680i)41-s + (0.733 − 0.680i)43-s + (−0.826 − 0.563i)47-s + (−0.623 + 0.781i)53-s + (−0.222 − 0.974i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4898280688 + 1.150542832i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4898280688 + 1.150542832i\) |
\(L(1)\) |
\(\approx\) |
\(0.9484969165 + 0.1494912109i\) |
\(L(1)\) |
\(\approx\) |
\(0.9484969165 + 0.1494912109i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.733 - 0.680i)T \) |
| 11 | \( 1 + (0.826 + 0.563i)T \) |
| 13 | \( 1 + (-0.0747 + 0.997i)T \) |
| 17 | \( 1 + (0.623 + 0.781i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.365 + 0.930i)T \) |
| 29 | \( 1 + (-0.365 + 0.930i)T \) |
| 31 | \( 1 + (-0.5 + 0.866i)T \) |
| 37 | \( 1 + (0.623 + 0.781i)T \) |
| 41 | \( 1 + (-0.733 - 0.680i)T \) |
| 43 | \( 1 + (0.733 - 0.680i)T \) |
| 47 | \( 1 + (-0.826 - 0.563i)T \) |
| 53 | \( 1 + (-0.623 + 0.781i)T \) |
| 59 | \( 1 + (-0.955 - 0.294i)T \) |
| 61 | \( 1 + (-0.365 + 0.930i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.623 - 0.781i)T \) |
| 73 | \( 1 + (0.900 + 0.433i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.0747 - 0.997i)T \) |
| 89 | \( 1 + (-0.900 - 0.433i)T \) |
| 97 | \( 1 + (0.5 + 0.866i)T \) |
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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
−19.83671137367128331863565458803, −18.95350903653621770665095026134, −18.45625576606100980236467110110, −17.68965368999388226327116900460, −16.69679464072916590717134296755, −16.11859845617338203575433900044, −15.291622621529222942751053215801, −14.57979453472515918664305290029, −14.03891886782024198980612791729, −13.02805572176373599013276223236, −12.201490847053138963849216034580, −11.39628627892852508705152822295, −10.99630255371177494272777481686, −9.89868729259018238858913987194, −9.29663196318355135164530486423, −8.05545238551793558197532540431, −7.718155741623564893069960278043, −6.72852938237861308856153446385, −5.97176174356266391450561549963, −5.018847198292431589489792227172, −3.97997552970734074545766557275, −3.23477388793726904538408610450, −2.5614692218011333503046352132, −1.04623189282711208191740892043, −0.264085272841736650086086855176,
1.20688242105391870910409786700, 1.6940247439468892009611860269, 3.29671747077481066376442794234, 3.867114332725816462580953173629, 4.788154293470230752008086321862, 5.487269341917555801669140102249, 6.6786309149457678342360010229, 7.33646998033621510395235679324, 8.12764725890647820371083455693, 9.13861056235894444072516026876, 9.44355453208339751828594072226, 10.631987052217126958162940781099, 11.53742880319330144344473131373, 12.09319797224078432031164855708, 12.6667368972642993331191080027, 13.69755489148827550074825330327, 14.420781626470284008923515209976, 15.210280143430936402860762533379, 15.91046573905222027287804356629, 16.81578941562106762530621229689, 17.062049540459150215417644618033, 18.21429215136588822635156886929, 18.95400333296154958683950103022, 19.77100892392689809694287035996, 20.09500546244262733115867646656