L(s) = 1 | + (−0.389 + 0.921i)2-s + (−0.587 + 0.809i)3-s + (−0.696 − 0.717i)4-s + (−0.516 − 0.856i)6-s + (0.931 − 0.362i)8-s + (−0.309 − 0.951i)9-s + (0.989 − 0.142i)12-s + (−0.441 − 0.897i)13-s + (−0.0285 + 0.999i)16-s + (0.676 − 0.736i)17-s + (0.996 + 0.0855i)18-s + (0.198 − 0.980i)19-s + (−0.281 − 0.959i)23-s + (−0.254 + 0.967i)24-s + (0.998 − 0.0570i)26-s + (0.951 + 0.309i)27-s + ⋯ |
L(s) = 1 | + (−0.389 + 0.921i)2-s + (−0.587 + 0.809i)3-s + (−0.696 − 0.717i)4-s + (−0.516 − 0.856i)6-s + (0.931 − 0.362i)8-s + (−0.309 − 0.951i)9-s + (0.989 − 0.142i)12-s + (−0.441 − 0.897i)13-s + (−0.0285 + 0.999i)16-s + (0.676 − 0.736i)17-s + (0.996 + 0.0855i)18-s + (0.198 − 0.980i)19-s + (−0.281 − 0.959i)23-s + (−0.254 + 0.967i)24-s + (0.998 − 0.0570i)26-s + (0.951 + 0.309i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7267294828 - 0.1357115886i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7267294828 - 0.1357115886i\) |
\(L(1)\) |
\(\approx\) |
\(0.6106076753 + 0.2498731861i\) |
\(L(1)\) |
\(\approx\) |
\(0.6106076753 + 0.2498731861i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.389 + 0.921i)T \) |
| 3 | \( 1 + (-0.587 + 0.809i)T \) |
| 13 | \( 1 + (-0.441 - 0.897i)T \) |
| 17 | \( 1 + (0.676 - 0.736i)T \) |
| 19 | \( 1 + (0.198 - 0.980i)T \) |
| 23 | \( 1 + (-0.281 - 0.959i)T \) |
| 29 | \( 1 + (0.870 + 0.491i)T \) |
| 31 | \( 1 + (0.985 - 0.170i)T \) |
| 37 | \( 1 + (-0.884 + 0.466i)T \) |
| 41 | \( 1 + (-0.974 + 0.226i)T \) |
| 43 | \( 1 + (-0.540 + 0.841i)T \) |
| 47 | \( 1 + (-0.996 + 0.0855i)T \) |
| 53 | \( 1 + (0.999 - 0.0285i)T \) |
| 59 | \( 1 + (0.974 + 0.226i)T \) |
| 61 | \( 1 + (0.921 - 0.389i)T \) |
| 67 | \( 1 + (0.755 + 0.654i)T \) |
| 71 | \( 1 + (0.993 - 0.113i)T \) |
| 73 | \( 1 + (-0.791 - 0.610i)T \) |
| 79 | \( 1 + (-0.774 - 0.633i)T \) |
| 83 | \( 1 + (0.825 + 0.564i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.967 - 0.254i)T \) |
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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
−18.59850311524337425162461982995, −17.69608698471115308251579068430, −17.34170376944104019058875326928, −16.63699245274317557590669974867, −16.07196721493592598413607156559, −14.86487398414554387186039040324, −13.84763268141046589738842158589, −13.72684828584506438190584056255, −12.66361087332268123388778513367, −12.10284023716742881209754219963, −11.77053718776420081481743930693, −11.01090261409228381075618771226, −10.02344868962454027838259766358, −9.895620571764411812845758386091, −8.501349233437138668331674784559, −8.246223194589580955410743800667, −7.2982651817927848700675706453, −6.70313810539326920016444474747, −5.66604148191567221774212023097, −5.039765519529808312090031940085, −4.05654775278283091887913570226, −3.34303135328319716775941414441, −2.24038738176419321901856626872, −1.715845154461543432688691151695, −0.93152533364864347811371412262,
0.34956559167172718896850312667, 1.10413867389804073417393986800, 2.65826120469134044317971949847, 3.46968196500859127213108098698, 4.5828216245093786228013704611, 5.00739108576729219309774829015, 5.57455367206919806957724824871, 6.58099130337283181165952968429, 6.92561850468630580269521158353, 8.06381009205636730304775004453, 8.553613781864671808076783537959, 9.444906485176391778464751386643, 10.11426608442249403981873663774, 10.39196960510351197212127846194, 11.42482207136208696113964464623, 12.08355978244902352978740464610, 12.99338393898820199060460237233, 13.785259330627215066170452817104, 14.61619564480862633275265264353, 15.03283256756848640055435476019, 15.86197519206405177688555849212, 16.174773541154897206552169291748, 16.96247708121380473264103307606, 17.56015120289410255062995610430, 18.027126584725320222601033221825