L(s) = 1 | + (−0.934 + 0.357i)2-s + (0.905 − 0.424i)3-s + (0.744 − 0.667i)4-s + (0.109 + 0.994i)5-s + (−0.694 + 0.719i)6-s + (0.989 − 0.145i)7-s + (−0.457 + 0.889i)8-s + (0.639 − 0.768i)9-s + (−0.457 − 0.889i)10-s + (0.520 + 0.853i)11-s + (0.391 − 0.920i)12-s + (−0.934 + 0.357i)13-s + (−0.872 + 0.489i)14-s + (0.520 + 0.853i)15-s + (0.109 − 0.994i)16-s + (−0.0365 + 0.999i)17-s + ⋯ |
L(s) = 1 | + (−0.934 + 0.357i)2-s + (0.905 − 0.424i)3-s + (0.744 − 0.667i)4-s + (0.109 + 0.994i)5-s + (−0.694 + 0.719i)6-s + (0.989 − 0.145i)7-s + (−0.457 + 0.889i)8-s + (0.639 − 0.768i)9-s + (−0.457 − 0.889i)10-s + (0.520 + 0.853i)11-s + (0.391 − 0.920i)12-s + (−0.934 + 0.357i)13-s + (−0.872 + 0.489i)14-s + (0.520 + 0.853i)15-s + (0.109 − 0.994i)16-s + (−0.0365 + 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 173 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.073765854 + 0.3169267141i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.073765854 + 0.3169267141i\) |
\(L(1)\) |
\(\approx\) |
\(1.014736750 + 0.1821929477i\) |
\(L(1)\) |
\(\approx\) |
\(1.014736750 + 0.1821929477i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 173 | \( 1 \) |
good | 2 | \( 1 + (-0.934 + 0.357i)T \) |
| 3 | \( 1 + (0.905 - 0.424i)T \) |
| 5 | \( 1 + (0.109 + 0.994i)T \) |
| 7 | \( 1 + (0.989 - 0.145i)T \) |
| 11 | \( 1 + (0.520 + 0.853i)T \) |
| 13 | \( 1 + (-0.934 + 0.357i)T \) |
| 17 | \( 1 + (-0.0365 + 0.999i)T \) |
| 19 | \( 1 + (-0.872 - 0.489i)T \) |
| 23 | \( 1 + (0.520 - 0.853i)T \) |
| 29 | \( 1 + (-0.694 - 0.719i)T \) |
| 31 | \( 1 + (0.905 + 0.424i)T \) |
| 37 | \( 1 + (-0.872 - 0.489i)T \) |
| 41 | \( 1 + (0.989 - 0.145i)T \) |
| 43 | \( 1 + (0.744 + 0.667i)T \) |
| 47 | \( 1 + (-0.181 + 0.983i)T \) |
| 53 | \( 1 + (-0.997 + 0.0729i)T \) |
| 59 | \( 1 + (0.957 - 0.288i)T \) |
| 61 | \( 1 + (-0.0365 - 0.999i)T \) |
| 67 | \( 1 + (0.905 - 0.424i)T \) |
| 71 | \( 1 + (-0.694 - 0.719i)T \) |
| 73 | \( 1 + (-0.581 + 0.813i)T \) |
| 79 | \( 1 + (-0.181 - 0.983i)T \) |
| 83 | \( 1 + (-0.976 - 0.217i)T \) |
| 89 | \( 1 + (0.252 - 0.967i)T \) |
| 97 | \( 1 + (-0.976 + 0.217i)T \) |
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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
−27.44800987920679798100109308662, −26.77484915287163810598076850754, −25.44136946171893387065487352294, −24.75295073288096623296157438781, −24.173714053627988380876648941475, −22.05369560920246963697979068226, −21.12143365054414405245575489652, −20.634959382558388073239265001895, −19.65438800203945498660040822367, −18.881585827739168094567331198, −17.50883396019418771831377002503, −16.73928726808991366899813896676, −15.730819069429031206071970593860, −14.6380627741555043386314901144, −13.42712381553309675361997465465, −12.20056691564425581480280580307, −11.15464152434361218317401812802, −9.89418849351226493842487376762, −8.96621708485764665238152858062, −8.32179253128400301539602950585, −7.34557542149862954222867268998, −5.31255615614524911805439422126, −4.01042916634488830497586997865, −2.55438092630868643902372849205, −1.33363151263977587944482434726,
1.73173807851816686335261864840, 2.52266701678701534662336658654, 4.39069140132646770241336598694, 6.37853405421362535722027206857, 7.19931835105016412221482842781, 8.02555632224962470382321994362, 9.153274707181696446171319834752, 10.18662750175946659331742785234, 11.21422360591992027070500532558, 12.5116088479564117830199437913, 14.304342627366265275516248073747, 14.64516936872778749305619842250, 15.429394992076656428198031668280, 17.36906867056671087717969222886, 17.614650454764159682430752012268, 18.99746638922109253851174583630, 19.3780704008136632093532066608, 20.56244348547137322645254635634, 21.47552653299846581614062697034, 23.0366391508836304187561995737, 24.17976780699505237343214761075, 24.8074328084943064802645030565, 25.883981825966639635625281358861, 26.45448696337928225092901639048, 27.27184181913310885295760224156