L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + i·5-s + (−0.866 + 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s − 12-s + 14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − i·18-s + (−0.866 + 0.5i)19-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + i·5-s + (−0.866 + 0.5i)6-s + (0.866 − 0.5i)7-s + i·8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s − 12-s + 14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s − i·18-s + (−0.866 + 0.5i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.533 + 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7511986968 + 1.361142335i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7511986968 + 1.361142335i\) |
\(L(1)\) |
\(\approx\) |
\(1.105668895 + 0.9566532758i\) |
\(L(1)\) |
\(\approx\) |
\(1.105668895 + 0.9566532758i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + iT \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (0.866 + 0.5i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (-0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.866 - 0.5i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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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
−28.13743862973901199994931414561, −27.66333119465544432054492651082, −25.473105186213034793779913817968, −24.5741469655824578227122107351, −23.95126100904154405283675605829, −23.274005296383212056116858819946, −21.88494096739672544101612036502, −21.20112246556485774518827309134, −19.97785792378785963379601052071, −19.20665651025219210423100566114, −17.93250991932621089960051760753, −16.96703982305744332135206367002, −15.631778674873224776175111463379, −14.48317917298045830735248191609, −13.2145025373012197428517371441, −12.65828426369246167922461082586, −11.625599716459771061605052190898, −10.85559530387101596575883746527, −9.05336071604074884867907321938, −7.80249596635769350576936093543, −6.27461018394639198914792996883, −5.31253812293272435097544973000, −4.36622109584558286458950090450, −2.30440222746446904192245962956, −1.270278108878699823518084427898,
2.62402277341173336311851569492, 3.984260005985334222826248846422, 4.85769045029722602065852481395, 6.16900348808263440713445034, 7.14469140563633673048930345747, 8.52003963956287163504501285977, 10.311207932740239631931394671137, 11.12600531294466617343083647107, 11.98717683630425313685162862720, 13.6369525380797279199890079836, 14.61210457084724168150945290800, 15.19647649952801421493145151249, 16.37798032535851026592052153378, 17.31932860060896554460203851730, 18.23489829259344682511690577033, 20.0781019789816287464151228408, 21.11416200325825154549775058858, 21.74939903932029855461743872538, 22.92881893427161708641231295082, 23.229398079095347072682702661532, 24.57375998668792028599808100287, 25.720851991339988460641956987914, 26.83166017257969484043713835888, 27.13897244667800716813947648431, 28.80847192678331121106385368201