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
−22.2696533678468911589460921378, −21.68910810170564781229113007425, −20.60302218510068414292979046199, −19.56102998531282874734526717393, −18.948136568020005536192166926935, −18.36347686264780995829098428233, −17.47459788406010922354274015171, −16.72599094016711731636672802742, −16.01674810924939248931172956452, −15.03279247848573768148440942156, −14.11098996913026170103698835545, −13.23709583077416970787028113393, −12.47679590994900703764127984026, −11.719233876743943292409652906892, −11.2238681923113083409100405086, −10.00272117184241838785358247664, −9.00815811364228296062487411054, −8.37757948172137521715926848475, −6.98217699084335910950960724233, −6.57913561717384912652583637819, −5.72700603021006440664134428385, −4.73162710170232080437282391262, −3.634409853853315011759202369104, −2.20946207292198719160364038810, −1.609519798586369705286452451305,
0.14439006908118613817666338754, 1.42045057840368582546168168690, 3.12151764008926805468887519914, 3.94210126051846248884123292492, 4.57819250071300764948852635426, 5.86160934453339234663289596410, 6.42102189389002104656491487663, 7.388365300246073287417730583686, 8.68352370875705815771452221538, 9.36352834876309051904173707952, 10.39024959828127309687693763923, 10.901804916979862669185537489, 11.63052046684951362370732350880, 12.84757863942086550139064505118, 13.344045351981020464384709583352, 14.67196840605677966948272301931, 15.12609422647148731098508038624, 16.239832960629852424162089635236, 16.71422392537600273992597720802, 17.48361652194308080642917912204, 18.12915446360211351089090004274, 19.48152668565216914911872665771, 19.98582655923854265542188419640, 20.78765824439465650944137138972, 21.75091397750860264368638069453