L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s − i·6-s + (−0.5 − 0.866i)7-s − i·8-s + (−0.5 + 0.866i)9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + i·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s − i·6-s + (−0.5 − 0.866i)7-s − i·8-s + (−0.5 + 0.866i)9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + i·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09053571291 + 0.3609777886i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09053571291 + 0.3609777886i\) |
\(L(1)\) |
\(\approx\) |
\(0.5117635123 + 0.1549053842i\) |
\(L(1)\) |
\(\approx\) |
\(0.5117635123 + 0.1549053842i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.866 + 0.5i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 - iT \) |
| 29 | \( 1 + iT \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.866 - 0.5i)T \) |
| 67 | \( 1 + (0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.866 + 0.5i)T \) |
| 83 | \( 1 + (-0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 - iT \) |
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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
−34.9066528272182748422424713341, −34.2664252735897850896988000063, −32.06244533698453825966401095974, −31.60795876488632470076245681358, −29.78682647071996046851146011617, −28.646196199413833750062426602047, −27.54638552764291938485488694739, −26.133635499869826101349533419327, −25.13040913425457295541449611357, −24.13644713481389304125848592437, −23.126285210517339395797933289601, −20.71377993510570818896521313163, −19.41528793437835292556803549603, −18.82854987735978636697766598597, −17.42523841987185044704441592555, −15.825254312647143323758725999466, −14.91450952323888946058735694712, −12.94725202815197280802513293547, −11.68970330135087068399409197331, −9.57824227482554863985948981713, −8.259581094013353617601059207090, −7.37105026768783361230159224211, −5.58252641741673014664747258603, −2.579042834589445045057661803713, −0.28571526045412949858901243672,
2.85855265856409634715555517549, 4.142512282580021924346953429121, 7.22402228956581453922790526349, 8.401096949666111721503734398547, 10.124678819231345667460929064096, 10.74968617119897751410260966260, 12.52926978284387350920693022684, 14.52280935518386128980969109728, 15.96938847173357718457398406582, 16.855246086112261092638360446, 18.81644238302818460813813070232, 19.68252818987094256134887775834, 20.7445538228875487608040231283, 22.00430910090601822485661219181, 23.4718722596492762351504810891, 25.5645122581727911814006034141, 26.52232107219536141045531880564, 27.08287362332175127631816440008, 28.37755120256851534303190719761, 29.75197791992480951706470429763, 30.98910836944265197884761088301, 32.03900869609211142328223506665, 33.73649918244985268010184616003, 34.60845765912475310626563227203, 36.20746601875277781328829846227