L(s) = 1 | + (0.989 − 0.142i)3-s + (−0.755 − 0.654i)7-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.755 + 0.654i)13-s + (−0.909 − 0.415i)17-s + (−0.415 − 0.909i)19-s + (−0.841 − 0.540i)21-s + (0.909 − 0.415i)27-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (0.755 − 0.654i)33-s + (−0.281 − 0.959i)37-s + (−0.654 + 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯ |
L(s) = 1 | + (0.989 − 0.142i)3-s + (−0.755 − 0.654i)7-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (−0.755 + 0.654i)13-s + (−0.909 − 0.415i)17-s + (−0.415 − 0.909i)19-s + (−0.841 − 0.540i)21-s + (0.909 − 0.415i)27-s + (0.415 − 0.909i)29-s + (0.142 − 0.989i)31-s + (0.755 − 0.654i)33-s + (−0.281 − 0.959i)37-s + (−0.654 + 0.755i)39-s + (−0.959 − 0.281i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0406 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0406 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.239252265 - 1.189819821i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239252265 - 1.189819821i\) |
\(L(1)\) |
\(\approx\) |
\(1.260827052 - 0.3760410684i\) |
\(L(1)\) |
\(\approx\) |
\(1.260827052 - 0.3760410684i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.989 - 0.142i)T \) |
| 7 | \( 1 + (-0.755 - 0.654i)T \) |
| 11 | \( 1 + (0.841 - 0.540i)T \) |
| 13 | \( 1 + (-0.755 + 0.654i)T \) |
| 17 | \( 1 + (-0.909 - 0.415i)T \) |
| 19 | \( 1 + (-0.415 - 0.909i)T \) |
| 29 | \( 1 + (0.415 - 0.909i)T \) |
| 31 | \( 1 + (0.142 - 0.989i)T \) |
| 37 | \( 1 + (-0.281 - 0.959i)T \) |
| 41 | \( 1 + (-0.959 - 0.281i)T \) |
| 43 | \( 1 + (0.989 - 0.142i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.755 + 0.654i)T \) |
| 59 | \( 1 + (0.654 + 0.755i)T \) |
| 61 | \( 1 + (0.142 - 0.989i)T \) |
| 67 | \( 1 + (0.540 - 0.841i)T \) |
| 71 | \( 1 + (-0.841 - 0.540i)T \) |
| 73 | \( 1 + (-0.909 + 0.415i)T \) |
| 79 | \( 1 + (-0.654 - 0.755i)T \) |
| 83 | \( 1 + (-0.281 - 0.959i)T \) |
| 89 | \( 1 + (0.142 + 0.989i)T \) |
| 97 | \( 1 + (-0.281 + 0.959i)T \) |
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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
−22.08670213557647225727374676206, −21.31995585763044108810325795844, −20.27195813272533070914854253313, −19.740615356119942024907229279371, −19.16272829640591825090288353336, −18.258377329292394703279324992732, −17.35835817863304398360286429154, −16.366751141062192042023639349460, −15.531732837233788642966948709785, −14.88733567240051309859702904930, −14.284973843305142874326416752466, −13.14819295451232396503891947751, −12.588608020783663013000421353341, −11.80965861705733043837279178569, −10.2952694500580809673837673455, −9.93171430106197712279024543157, −8.83352305381714301988856641538, −8.44386534444148662997335155616, −7.15544024201883535161874668515, −6.55228325928686994871699607726, −5.296863162080463617121991686295, −4.248270098680575966250090799931, −3.358590946328343081213068741656, −2.49058462781167367993205494901, −1.55201144406058459155163667628,
0.64497563433690816458440676761, 2.07494091272834784074189167584, 2.882962570978622124071030753397, 3.984999199794983142477130425419, 4.506075677855423957350304093612, 6.183041672832162316530258254769, 6.91597213530211232981252868791, 7.56458270206384213348514280047, 8.827102220518205836578708092942, 9.26481241934172251035598850967, 10.097605250988251154007924798729, 11.159720857066445206525892262610, 12.120634551489905188937882845344, 13.109405527693897606351041248122, 13.70818539538834184019576822622, 14.33290890812079587582336393990, 15.29837923901961077770832883651, 16.021074991111959716941070703483, 16.95350652844703592022378069004, 17.6634941500510760818915911796, 18.97512575668361559381469553110, 19.333400866758648997875301159954, 19.97018840921275675557590744381, 20.74069217656867785226308660058, 21.748652931139329590526107013478