L(s) = 1 | + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)5-s + (0.222 + 0.974i)8-s + (−0.222 + 0.974i)10-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)13-s + (−0.222 + 0.974i)16-s + (−0.623 + 0.781i)17-s + 19-s + (−0.623 + 0.781i)20-s + (0.623 + 0.781i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.623 − 0.781i)26-s + ⋯ |
L(s) = 1 | + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)5-s + (0.222 + 0.974i)8-s + (−0.222 + 0.974i)10-s + (0.900 + 0.433i)11-s + (−0.900 − 0.433i)13-s + (−0.222 + 0.974i)16-s + (−0.623 + 0.781i)17-s + 19-s + (−0.623 + 0.781i)20-s + (0.623 + 0.781i)22-s + (−0.623 − 0.781i)23-s + (−0.900 + 0.433i)25-s + (−0.623 − 0.781i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.599092409 + 2.637870187i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.599092409 + 2.637870187i\) |
\(L(1)\) |
\(\approx\) |
\(1.563559173 + 1.035782741i\) |
\(L(1)\) |
\(\approx\) |
\(1.563559173 + 1.035782741i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (0.222 + 0.974i)T \) |
| 11 | \( 1 + (0.900 + 0.433i)T \) |
| 13 | \( 1 + (-0.900 - 0.433i)T \) |
| 17 | \( 1 + (-0.623 + 0.781i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.623 - 0.781i)T \) |
| 29 | \( 1 + (-0.623 + 0.781i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (0.623 - 0.781i)T \) |
| 41 | \( 1 + (0.222 + 0.974i)T \) |
| 43 | \( 1 + (-0.222 + 0.974i)T \) |
| 47 | \( 1 + (0.900 + 0.433i)T \) |
| 53 | \( 1 + (-0.623 - 0.781i)T \) |
| 59 | \( 1 + (0.222 - 0.974i)T \) |
| 61 | \( 1 + (0.623 - 0.781i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.623 - 0.781i)T \) |
| 73 | \( 1 + (-0.900 + 0.433i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.900 - 0.433i)T \) |
| 97 | \( 1 + T \) |
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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
−27.87370044240249022677323495010, −26.82098930755348509028332906753, −25.14964454322502891947051110262, −24.50867783475692968306399099488, −23.81202722448504634526681231149, −22.4181561903771655826744743045, −21.81364505655487693049800095993, −20.64843034784239871200042912250, −19.95407151023763281640587528961, −18.996079395633952665121246704584, −17.410494952946015075093495696923, −16.37153733375941680059852993283, −15.391241675677682641475412494835, −14.00777734571314246394912655624, −13.46299205810516515137791369237, −12.03670939432252260117042999486, −11.63449423730782672630068761732, −9.92162366874562518293141493981, −9.09479218090493271257387196566, −7.34900543339697770699731678312, −5.9906478365915661610525353133, −4.92881443550561150559856270055, −3.89610059498076170343511855707, −2.28164873742631371571846383781, −0.88902845351441869281264737730,
2.13215132085697260312881748665, 3.35133030058109312833687895019, 4.60739527175773689483396819990, 6.03322855497322928585346983906, 6.89478807223548274418031475706, 7.93798419996321890755599435580, 9.64042186801856329432010624560, 10.92880233073259159422421575380, 11.97973219429765002752904966270, 13.04009046107538050876126820498, 14.35984834761135722090739922502, 14.74748022795431880178249241144, 15.91821241862510170499205915320, 17.19424808555266783696760075620, 17.96519656286569656748430547277, 19.51013160315279625155828438937, 20.40393495411929336547228283423, 21.903332538340154123107508824138, 22.21289167080335948669831258196, 23.157770350978658679923629756984, 24.4224949487360628217415174413, 25.097151402420817223797027009848, 26.2415082184702560289290699746, 26.86734051404304801258022167235, 28.41707450800003443583669212024