L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.415 + 0.909i)3-s + (−0.959 + 0.281i)4-s + (−0.959 − 0.281i)6-s + (−0.841 + 0.540i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.142 − 0.989i)12-s + (−0.841 − 0.540i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (0.959 + 0.281i)17-s + (0.654 − 0.755i)18-s + (−0.959 + 0.281i)19-s + ⋯ |
L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.415 + 0.909i)3-s + (−0.959 + 0.281i)4-s + (−0.959 − 0.281i)6-s + (−0.841 + 0.540i)7-s + (−0.415 − 0.909i)8-s + (−0.654 − 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.142 − 0.989i)12-s + (−0.841 − 0.540i)13-s + (−0.654 − 0.755i)14-s + (0.841 − 0.540i)16-s + (0.959 + 0.281i)17-s + (0.654 − 0.755i)18-s + (−0.959 + 0.281i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.763 - 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1737697020 + 0.4744073259i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1737697020 + 0.4744073259i\) |
\(L(1)\) |
\(\approx\) |
\(0.3655983593 + 0.5648328054i\) |
\(L(1)\) |
\(\approx\) |
\(0.3655983593 + 0.5648328054i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.841 + 0.540i)T \) |
| 11 | \( 1 + (-0.142 + 0.989i)T \) |
| 13 | \( 1 + (-0.841 - 0.540i)T \) |
| 17 | \( 1 + (0.959 + 0.281i)T \) |
| 19 | \( 1 + (-0.959 + 0.281i)T \) |
| 29 | \( 1 + (-0.959 - 0.281i)T \) |
| 31 | \( 1 + (0.415 + 0.909i)T \) |
| 37 | \( 1 + (0.654 + 0.755i)T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + (-0.415 + 0.909i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (-0.841 + 0.540i)T \) |
| 59 | \( 1 + (0.841 + 0.540i)T \) |
| 61 | \( 1 + (0.415 + 0.909i)T \) |
| 67 | \( 1 + (0.142 + 0.989i)T \) |
| 71 | \( 1 + (-0.142 - 0.989i)T \) |
| 73 | \( 1 + (0.959 - 0.281i)T \) |
| 79 | \( 1 + (0.841 + 0.540i)T \) |
| 83 | \( 1 + (0.654 + 0.755i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
| 97 | \( 1 + (0.654 - 0.755i)T \) |
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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
−29.03422639432898916788090851358, −27.98177064886252402939388126882, −26.81732468610646606362843612202, −25.71870072167066909853987683210, −24.20479719123811144858550093615, −23.46116027160754739830336312976, −22.50362174128530211120514363718, −21.59629626490291203239624812586, −20.23452078780113598254409321582, −19.08702050167291933966900396294, −18.86110938681255219161582020244, −17.306205854884359344971841795327, −16.52850205180166255560588936234, −14.4678002781719172115118497329, −13.4757697023180688555879249396, −12.696421904960526181566728546520, −11.647932489145365215042368362650, −10.63291199469799348982001650010, −9.399592244960054538594066160, −7.95004242825062538050822115583, −6.515247531156862128892888865517, −5.2492721155096988228333109384, −3.5750096551382545313452001800, −2.21333934851078255977198639131, −0.47558893852744344433186668546,
3.16859264678050938064899146447, 4.552929029078934027361505056759, 5.588883945366500606863540564003, 6.66183388275861022440370953755, 8.15461655604027509192788652825, 9.5531060010117707021057253274, 10.12883862142128522372320169790, 12.10615430259548954171339808560, 12.93557103605278499517401628236, 14.76447561566892746881576616893, 15.15312314541193174765563285138, 16.339300644782667354249413272707, 17.08712384856090488750259714382, 18.16328649987554478305568306103, 19.51233554727832059644640005801, 21.02535200837310346450670231830, 22.0441815803955969058140375624, 22.78812810717119679610621968064, 23.56456171506906329011964647672, 25.1369333475357650818278596128, 25.72617872601928320488017640698, 26.77569064798373578517849717849, 27.77951085883021788081186618819, 28.47749777361771353977882995816, 29.91251445117234843757131223693