L(s) = 1 | + (0.0747 − 0.997i)5-s + (−0.365 + 0.930i)11-s + (−0.988 + 0.149i)13-s + (−0.222 − 0.974i)17-s − 19-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.733 + 0.680i)29-s + (0.5 − 0.866i)31-s + (−0.222 − 0.974i)37-s + (0.0747 − 0.997i)41-s + (−0.0747 − 0.997i)43-s + (−0.365 + 0.930i)47-s + (−0.222 + 0.974i)53-s + (0.900 + 0.433i)55-s + ⋯ |
L(s) = 1 | + (0.0747 − 0.997i)5-s + (−0.365 + 0.930i)11-s + (−0.988 + 0.149i)13-s + (−0.222 − 0.974i)17-s − 19-s + (0.733 + 0.680i)23-s + (−0.988 − 0.149i)25-s + (−0.733 + 0.680i)29-s + (0.5 − 0.866i)31-s + (−0.222 − 0.974i)37-s + (0.0747 − 0.997i)41-s + (−0.0747 − 0.997i)43-s + (−0.365 + 0.930i)47-s + (−0.222 + 0.974i)53-s + (0.900 + 0.433i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.130418158 + 0.2369131474i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.130418158 + 0.2369131474i\) |
\(L(1)\) |
\(\approx\) |
\(0.8770335039 - 0.1094889637i\) |
\(L(1)\) |
\(\approx\) |
\(0.8770335039 - 0.1094889637i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.0747 - 0.997i)T \) |
| 11 | \( 1 + (-0.365 + 0.930i)T \) |
| 13 | \( 1 + (-0.988 + 0.149i)T \) |
| 17 | \( 1 + (-0.222 - 0.974i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (0.733 + 0.680i)T \) |
| 29 | \( 1 + (-0.733 + 0.680i)T \) |
| 31 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.222 - 0.974i)T \) |
| 41 | \( 1 + (0.0747 - 0.997i)T \) |
| 43 | \( 1 + (-0.0747 - 0.997i)T \) |
| 47 | \( 1 + (-0.365 + 0.930i)T \) |
| 53 | \( 1 + (-0.222 + 0.974i)T \) |
| 59 | \( 1 + (-0.826 + 0.563i)T \) |
| 61 | \( 1 + (-0.733 + 0.680i)T \) |
| 67 | \( 1 + (0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.623 - 0.781i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.988 + 0.149i)T \) |
| 89 | \( 1 + (0.623 - 0.781i)T \) |
| 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
−19.803518205650303691241391724038, −19.12463398528010660268622324880, −18.75598014180903137198474291279, −17.769619560922927648427599864069, −17.119240469828549308851668795890, −16.424125199210467718175520182911, −15.29212526138817688976172732616, −14.911314475533829362747575752560, −14.18604984440886499873390873497, −13.27862074153640579844085119596, −12.66045636148633677455428237269, −11.59981750369321183544446018151, −10.912465581091159237122425649420, −10.33005313453185727955704671013, −9.56404454432281315801430913386, −8.396824868329725807952686802216, −7.92812441982941486027750915045, −6.73420515450982857438289651965, −6.37098259711912014356237528290, −5.33573946682778777771434432951, −4.39380256019673038834465484284, −3.338817473878261301308084692717, −2.688649960115770728728529348471, −1.767207132308378319250397214595, −0.31505089310005724377591326234,
0.58628792159709780302574676897, 1.83551527766108768631777465275, 2.4802278147974505983000218713, 3.80279566856673688524783128420, 4.76998422100867123927801679143, 5.1039378468011576601645036403, 6.188764738866609330815249879523, 7.34991961649221102355361290071, 7.68291365759707276818005228045, 9.06984387863144675996139817091, 9.2352891426337294716022435573, 10.2284867598513456318956327785, 11.10519255421348510381786614009, 12.146715332910563992787010097818, 12.52120746084150672601021636837, 13.32220543036650752074064361951, 14.08447749085622475342713326252, 15.12786331694056791557211377143, 15.54887796520327181251446869704, 16.590047508638877429369808933092, 17.12135862774162722480492322226, 17.7293034625632087225196112372, 18.662368252526090760825692058142, 19.526600593759827211321012479870, 20.10633268243335269267449289449