L(s) = 1 | + (0.736 + 0.676i)2-s + (0.809 − 0.587i)3-s + (0.0855 + 0.996i)4-s + (0.0285 + 0.999i)5-s + (0.993 + 0.113i)6-s + (−0.610 + 0.791i)8-s + (0.309 − 0.951i)9-s + (−0.654 + 0.755i)10-s + (0.654 + 0.755i)12-s + (−0.921 + 0.389i)13-s + (0.610 + 0.791i)15-s + (−0.985 + 0.170i)16-s + (−0.254 + 0.967i)17-s + (0.870 − 0.491i)18-s + (−0.362 + 0.931i)19-s + (−0.993 + 0.113i)20-s + ⋯ |
L(s) = 1 | + (0.736 + 0.676i)2-s + (0.809 − 0.587i)3-s + (0.0855 + 0.996i)4-s + (0.0285 + 0.999i)5-s + (0.993 + 0.113i)6-s + (−0.610 + 0.791i)8-s + (0.309 − 0.951i)9-s + (−0.654 + 0.755i)10-s + (0.654 + 0.755i)12-s + (−0.921 + 0.389i)13-s + (0.610 + 0.791i)15-s + (−0.985 + 0.170i)16-s + (−0.254 + 0.967i)17-s + (0.870 − 0.491i)18-s + (−0.362 + 0.931i)19-s + (−0.993 + 0.113i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.145618835 + 2.191828715i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145618835 + 2.191828715i\) |
\(L(1)\) |
\(\approx\) |
\(1.509822994 + 0.9833237734i\) |
\(L(1)\) |
\(\approx\) |
\(1.509822994 + 0.9833237734i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.736 + 0.676i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.0285 + 0.999i)T \) |
| 13 | \( 1 + (-0.921 + 0.389i)T \) |
| 17 | \( 1 + (-0.254 + 0.967i)T \) |
| 19 | \( 1 + (-0.362 + 0.931i)T \) |
| 23 | \( 1 + (-0.142 + 0.989i)T \) |
| 29 | \( 1 + (0.998 + 0.0570i)T \) |
| 31 | \( 1 + (-0.516 - 0.856i)T \) |
| 37 | \( 1 + (0.974 - 0.226i)T \) |
| 41 | \( 1 + (0.198 + 0.980i)T \) |
| 43 | \( 1 + (0.959 - 0.281i)T \) |
| 47 | \( 1 + (0.870 + 0.491i)T \) |
| 53 | \( 1 + (-0.985 - 0.170i)T \) |
| 59 | \( 1 + (-0.198 + 0.980i)T \) |
| 61 | \( 1 + (-0.736 + 0.676i)T \) |
| 67 | \( 1 + (0.415 - 0.909i)T \) |
| 71 | \( 1 + (0.774 + 0.633i)T \) |
| 73 | \( 1 + (0.696 - 0.717i)T \) |
| 79 | \( 1 + (0.564 - 0.825i)T \) |
| 83 | \( 1 + (0.897 - 0.441i)T \) |
| 89 | \( 1 + (-0.841 - 0.540i)T \) |
| 97 | \( 1 + (0.0285 - 0.999i)T \) |
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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
−21.76176953772312615897308185564, −20.958560109287667348998039426826, −20.25516654185316348153657828016, −19.85534096932247267920818594365, −19.09686300996296063069305588916, −17.962678553458227013266337534988, −16.84794032381007792712349106026, −15.85884263974902564406731592410, −15.40731682323949701563260738920, −14.31051088490522216982416619466, −13.83672181205709921951440055122, −12.81477566561188707509063317853, −12.35533022610107728363142179110, −11.19044944488901542643430480062, −10.34228203571233066781843508150, −9.4504924036737629509652037342, −8.94701902684050481812857749060, −7.84688021044285697853909932036, −6.66408517017119877333161763144, −5.22701939014351834781399907432, −4.79145936878621185106118552985, −4.01594010339484556467447252729, −2.79084831532007568598048237272, −2.20690882636900363031257489589, −0.7385421281789049090982459569,
1.90103176391968029513055563035, 2.684483743129842152871019349474, 3.618482060154548603451816602447, 4.38820035181730403251191841338, 5.95106505913423491707596073569, 6.42260358382092027441327482970, 7.55032824150292709907595732630, 7.77647384679496171446058513045, 8.9883691927550168450054458949, 9.93817519892806808861342999846, 11.16006531494953343357687977116, 12.13185331117631337533015402268, 12.802171348273802202027042121661, 13.750583707607931309589829397622, 14.35798241942731687496631617938, 14.94484692144369716990656198346, 15.55630439198298621278142938275, 16.786077176510911934171773993607, 17.60127926737656903483079547133, 18.33979762523014486172558542765, 19.270440559314947730161835558341, 19.885148843108898124960359055575, 21.07557524313602756304827860916, 21.6454055311322070598833595508, 22.41433612742652311509696394268