L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.222 + 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)8-s + (0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s + 17-s + (0.900 − 0.433i)19-s + (−0.623 − 0.781i)20-s + (−0.900 − 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 + 0.433i)4-s + (0.222 + 0.974i)5-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)8-s + (0.900 − 0.433i)10-s + (0.623 − 0.781i)11-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)14-s + (0.623 − 0.781i)16-s + 17-s + (0.900 − 0.433i)19-s + (−0.623 − 0.781i)20-s + (−0.900 − 0.433i)22-s + (0.222 − 0.974i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0833 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0833 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9681194684 - 0.8905022645i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9681194684 - 0.8905022645i\) |
\(L(1)\) |
\(\approx\) |
\(0.8566685111 - 0.4167359980i\) |
\(L(1)\) |
\(\approx\) |
\(0.8566685111 - 0.4167359980i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (0.222 + 0.974i)T \) |
| 7 | \( 1 + (-0.900 - 0.433i)T \) |
| 11 | \( 1 + (0.623 - 0.781i)T \) |
| 13 | \( 1 + (0.623 - 0.781i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.900 - 0.433i)T \) |
| 23 | \( 1 + (0.222 - 0.974i)T \) |
| 31 | \( 1 + (0.222 + 0.974i)T \) |
| 37 | \( 1 + (-0.623 - 0.781i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (0.222 - 0.974i)T \) |
| 47 | \( 1 + (0.623 - 0.781i)T \) |
| 53 | \( 1 + (0.222 + 0.974i)T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + (0.900 + 0.433i)T \) |
| 67 | \( 1 + (0.623 + 0.781i)T \) |
| 71 | \( 1 + (-0.623 + 0.781i)T \) |
| 73 | \( 1 + (0.222 - 0.974i)T \) |
| 79 | \( 1 + (-0.623 - 0.781i)T \) |
| 83 | \( 1 + (0.900 - 0.433i)T \) |
| 89 | \( 1 + (-0.222 - 0.974i)T \) |
| 97 | \( 1 + (0.900 - 0.433i)T \) |
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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
−31.06426352463631670176825485936, −29.264155959417921831522459189335, −28.24220766581954054625595816183, −27.59902471878582372667215743592, −26.030715150940449580367853225906, −25.37374373897941544613082296483, −24.47505824016770229512872822542, −23.33147418583224973152116588372, −22.39983094142114865959072622921, −21.0450368206603942188511555519, −19.64282235685026065599432032102, −18.61798113601364909704552386058, −17.31491511404049413985800180382, −16.42166521562045052326593638827, −15.61318518946571758660181589369, −14.19844193154742436140330928044, −13.10265804713773313584069946442, −11.995080744706986609328066098512, −9.69689332674239998564622249758, −9.26434383070295816102923496199, −7.8313522639844138314448227272, −6.41155240558150899371144767951, −5.35102184383736455184793185054, −3.90591653710958531089955909786, −1.26157496029690552368255135222,
0.82526001673522256413569574887, 2.90540017604993240544233158599, 3.67395229681909455007986958979, 5.793404275196831072622109931668, 7.294486530539534207361603928311, 8.88649098234636779571008372979, 10.15124372348524244879943054349, 10.87497186362288820801824141823, 12.20940444883336492163524200201, 13.50434285116375131111478059157, 14.31682584243763705812371159008, 16.08682941867964825228108828344, 17.36249420638555996194232475380, 18.51963663879309384091936890670, 19.279948821302866141267563490414, 20.34870836658231892841934371156, 21.62222531739256837144835840572, 22.53409178198073422675890780324, 23.20082425763678106905175367340, 25.14191997051667815141836418628, 26.255071995222847133707620134281, 26.924950557948294663172343211027, 28.14802041924235257331193000731, 29.32560003500730695774027521426, 29.97970246917994436345140463746