L(s) = 1 | + (0.669 + 0.743i)2-s + (−0.5 − 0.866i)3-s + (−0.104 + 0.994i)4-s + (−0.913 + 0.406i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.5 + 0.866i)9-s + (−0.913 − 0.406i)10-s + (−0.913 − 0.406i)11-s + (0.913 − 0.406i)12-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.978 − 0.207i)16-s + (0.913 + 0.406i)17-s + (−0.978 + 0.207i)18-s + (−0.978 − 0.207i)19-s + ⋯ |
L(s) = 1 | + (0.669 + 0.743i)2-s + (−0.5 − 0.866i)3-s + (−0.104 + 0.994i)4-s + (−0.913 + 0.406i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + (−0.5 + 0.866i)9-s + (−0.913 − 0.406i)10-s + (−0.913 − 0.406i)11-s + (0.913 − 0.406i)12-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (−0.978 − 0.207i)16-s + (0.913 + 0.406i)17-s + (−0.978 + 0.207i)18-s + (−0.978 − 0.207i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.988 - 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.327129608 - 0.09939944074i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.327129608 - 0.09939944074i\) |
\(L(1)\) |
\(\approx\) |
\(0.9653128245 + 0.1928873421i\) |
\(L(1)\) |
\(\approx\) |
\(0.9653128245 + 0.1928873421i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 41 | \( 1 \) |
good | 2 | \( 1 + (0.669 + 0.743i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.913 + 0.406i)T \) |
| 11 | \( 1 + (-0.913 - 0.406i)T \) |
| 13 | \( 1 + (0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.913 + 0.406i)T \) |
| 19 | \( 1 + (-0.978 - 0.207i)T \) |
| 23 | \( 1 + (0.669 + 0.743i)T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.913 - 0.406i)T \) |
| 37 | \( 1 + (0.913 - 0.406i)T \) |
| 43 | \( 1 + (0.309 - 0.951i)T \) |
| 47 | \( 1 + (0.669 + 0.743i)T \) |
| 53 | \( 1 + (0.104 - 0.994i)T \) |
| 59 | \( 1 + (0.978 - 0.207i)T \) |
| 61 | \( 1 + (0.978 + 0.207i)T \) |
| 67 | \( 1 + (0.104 - 0.994i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + (-0.978 - 0.207i)T \) |
| 97 | \( 1 + (-0.809 - 0.587i)T \) |
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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
−25.25878361999024343636201647945, −23.77017128377991618759873538312, −23.40806745868709889753305190122, −22.73949169146349376101731612464, −21.496930999970442142562264067246, −20.95738764810982743527672777666, −20.22286128567731524305661489579, −19.11853547245284915982396715641, −18.29299215309718503031391076733, −16.7811321387965860950503697084, −16.00737883733138454518895402040, −15.14597432440666233353075704647, −14.3584154121930955444673059120, −12.92039129032729249828037227573, −12.151393148534800900117097499564, −11.3046191261799776179003729442, −10.52012475753192577990053540897, −9.52001143622802986525018269063, −8.43159784672166477628494398330, −6.80226676908012727175634056687, −5.50486181906532397880293124594, −4.59526970896918887525342062033, −3.93135877582062801602026490968, −2.68426973065602677783149471144, −0.84634162199206274264192988460,
0.48437770464536670591273459889, 2.620990425475438741541937198217, 3.65321741303406633205452811046, 5.12299567614543117027802047751, 5.93109748304310340914306203517, 7.054476149538620286630748451949, 7.8306266293894787037961207573, 8.47553363402751505909616591396, 10.64042298329818460116386215303, 11.39309213104341299329309698353, 12.58621841990720289603556773851, 12.99104667692850193823342588834, 14.20003549702928880430261649928, 15.16947973251635139653784970788, 16.00912161107005284083898960094, 16.93132977046219903814233814244, 17.937169439471592986579258811468, 18.72019694457597942341956284686, 19.69159443693133371687738693167, 20.973092472193965227537715114306, 22.03613813461569832565293473495, 22.93742200822588549503231152996, 23.63994955153811786545218969844, 23.90819481685339388839323775737, 25.335123161240442284639778076289