L(s) = 1 | + (−0.654 − 0.755i)3-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.841 − 0.540i)17-s + (−0.841 + 0.540i)19-s + (0.959 − 0.281i)21-s + (0.841 − 0.540i)27-s + (−0.841 − 0.540i)29-s + (−0.654 + 0.755i)31-s + (−0.415 − 0.909i)33-s + (−0.142 + 0.989i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
L(s) = 1 | + (−0.654 − 0.755i)3-s + (−0.415 + 0.909i)7-s + (−0.142 + 0.989i)9-s + (0.959 + 0.281i)11-s + (0.415 + 0.909i)13-s + (−0.841 − 0.540i)17-s + (−0.841 + 0.540i)19-s + (0.959 − 0.281i)21-s + (0.841 − 0.540i)27-s + (−0.841 − 0.540i)29-s + (−0.654 + 0.755i)31-s + (−0.415 − 0.909i)33-s + (−0.142 + 0.989i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09018151787 + 0.2874670975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09018151787 + 0.2874670975i\) |
\(L(1)\) |
\(\approx\) |
\(0.6751604392 + 0.01893405926i\) |
\(L(1)\) |
\(\approx\) |
\(0.6751604392 + 0.01893405926i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (-0.415 + 0.909i)T \) |
| 11 | \( 1 + (0.959 + 0.281i)T \) |
| 13 | \( 1 + (0.415 + 0.909i)T \) |
| 17 | \( 1 + (-0.841 - 0.540i)T \) |
| 19 | \( 1 + (-0.841 + 0.540i)T \) |
| 29 | \( 1 + (-0.841 - 0.540i)T \) |
| 31 | \( 1 + (-0.654 + 0.755i)T \) |
| 37 | \( 1 + (-0.142 + 0.989i)T \) |
| 41 | \( 1 + (-0.142 - 0.989i)T \) |
| 43 | \( 1 + (-0.654 - 0.755i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.415 - 0.909i)T \) |
| 61 | \( 1 + (0.654 - 0.755i)T \) |
| 67 | \( 1 + (-0.959 + 0.281i)T \) |
| 71 | \( 1 + (-0.959 + 0.281i)T \) |
| 73 | \( 1 + (-0.841 + 0.540i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (-0.142 + 0.989i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (0.142 + 0.989i)T \) |
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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
−21.75091397750860264368638069453, −20.78765824439465650944137138972, −19.98582655923854265542188419640, −19.48152668565216914911872665771, −18.12915446360211351089090004274, −17.48361652194308080642917912204, −16.71422392537600273992597720802, −16.239832960629852424162089635236, −15.12609422647148731098508038624, −14.67196840605677966948272301931, −13.344045351981020464384709583352, −12.84757863942086550139064505118, −11.63052046684951362370732350880, −10.901804916979862669185537489, −10.39024959828127309687693763923, −9.36352834876309051904173707952, −8.68352370875705815771452221538, −7.388365300246073287417730583686, −6.42102189389002104656491487663, −5.86160934453339234663289596410, −4.57819250071300764948852635426, −3.94210126051846248884123292492, −3.12151764008926805468887519914, −1.42045057840368582546168168690, −0.14439006908118613817666338754,
1.609519798586369705286452451305, 2.20946207292198719160364038810, 3.634409853853315011759202369104, 4.73162710170232080437282391262, 5.72700603021006440664134428385, 6.57913561717384912652583637819, 6.98217699084335910950960724233, 8.37757948172137521715926848475, 9.00815811364228296062487411054, 10.00272117184241838785358247664, 11.2238681923113083409100405086, 11.719233876743943292409652906892, 12.47679590994900703764127984026, 13.23709583077416970787028113393, 14.11098996913026170103698835545, 15.03279247848573768148440942156, 16.01674810924939248931172956452, 16.72599094016711731636672802742, 17.47459788406010922354274015171, 18.36347686264780995829098428233, 18.948136568020005536192166926935, 19.56102998531282874734526717393, 20.60302218510068414292979046199, 21.68910810170564781229113007425, 22.2696533678468911589460921378