L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s − 13-s + 14-s + 16-s + 17-s − 19-s − 20-s − 22-s − 23-s + 25-s + 26-s − 28-s + 29-s + 31-s − 32-s − 34-s + 35-s + 37-s + 38-s + 40-s + 43-s + ⋯ |
L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s + 11-s − 13-s + 14-s + 16-s + 17-s − 19-s − 20-s − 22-s − 23-s + 25-s + 26-s − 28-s + 29-s + 31-s − 32-s − 34-s + 35-s + 37-s + 38-s + 40-s + 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6945011863\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6945011863\) |
\(L(1)\) |
\(\approx\) |
\(0.5665357400\) |
\(L(1)\) |
\(\approx\) |
\(0.5665357400\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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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
−28.496527443601652408427865635740, −27.59917777227529689878228166703, −26.89432462694290690808266950409, −25.83509616929602627444056089226, −24.938416984263702470065988871, −23.826188122962838716435547624147, −22.727924817343928308288707222516, −21.5204251909043245028683405365, −19.997861758964299517454572338119, −19.49391290582832650661058388718, −18.75126776465894076802508446038, −17.23025221714950748742520443585, −16.46317566449174461431645527985, −15.50350436889921861898463172209, −14.4471050579784769911305083960, −12.40809531095054900728934171, −11.888783154224987558623500060129, −10.43212103150766259393218738374, −9.479978199540395501709374256384, −8.28458807072094486078474133589, −7.19396503957577878369835475155, −6.178193289599403082810949075623, −4.04990216085087181767171763133, −2.70064330904441363956042357960, −0.69365918792444668323623063934,
0.69365918792444668323623063934, 2.70064330904441363956042357960, 4.04990216085087181767171763133, 6.178193289599403082810949075623, 7.19396503957577878369835475155, 8.28458807072094486078474133589, 9.479978199540395501709374256384, 10.43212103150766259393218738374, 11.888783154224987558623500060129, 12.40809531095054900728934171, 14.4471050579784769911305083960, 15.50350436889921861898463172209, 16.46317566449174461431645527985, 17.23025221714950748742520443585, 18.75126776465894076802508446038, 19.49391290582832650661058388718, 19.997861758964299517454572338119, 21.5204251909043245028683405365, 22.727924817343928308288707222516, 23.826188122962838716435547624147, 24.938416984263702470065988871, 25.83509616929602627444056089226, 26.89432462694290690808266950409, 27.59917777227529689878228166703, 28.496527443601652408427865635740