L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s − 37-s − 39-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)9-s − 11-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s − 27-s + (0.5 − 0.866i)29-s − 31-s + (−0.5 − 0.866i)33-s − 37-s − 39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.3635165898 + 1.131922398i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.3635165898 + 1.131922398i\) |
\(L(1)\) |
\(\approx\) |
\(0.8858915522 + 0.6344422651i\) |
\(L(1)\) |
\(\approx\) |
\(0.8858915522 + 0.6344422651i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 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
−23.02672639254912121435210004341, −21.77493518432801468385427461049, −20.83802137684713199441438456213, −20.27981536309937113736312578274, −19.49984905050847206936989349772, −18.40645965399008065561368098660, −17.847036499022473431738837531357, −16.979419740302187557222530366867, −15.95901351845396900093459790137, −15.003501950098003186806855726748, −13.959575998795239091997685340688, −13.246202495189585725274477373996, −12.61513061341047387423991082919, −11.83201737873744616503125828750, −10.451678481220632331574388661097, −9.49307912550867847409525934754, −8.62889295273213064581263591074, −7.76715606775173429172613425258, −6.99491337539961118708070841055, −5.57020503024303049855258350567, −5.1007538036044224201796651539, −3.38257331787598000267090048654, −2.45164340007763903985395637805, −1.33171239296466862050103505078, −0.258780652773546407962305464357,
1.983505924597395711463993874993, 2.77730793472336224500808246166, 3.79255322795703829509008210119, 4.90176665873155955366521727111, 5.85792370786733437326299425146, 7.02034429596559183746589809766, 8.02200632213412731724727598944, 9.02723017876685244243513490702, 10.02059609699873892323789621241, 10.50797400351262033509409646561, 11.398932633532710689963384733890, 12.73761259946196952014995131261, 13.74571519869103350692671375616, 14.50367112782692981734860852362, 15.08399522128245996568164481513, 16.05965560966275858753467422480, 16.90912432791525108017732511140, 17.83264366674991313368537047820, 18.95930087821984378626715425726, 19.367477809853015971954006489013, 20.78171893253597779047234428435, 21.16091765774086329795545766315, 21.986965978499035517541915443653, 22.69643879998264902951533019714, 23.68228537243809159071211539630