L(s) = 1 | − 2.44i·5-s − 2i·7-s + 4.89·11-s − 3.46·13-s − 4.24i·17-s − 6.92i·19-s − 8.48·23-s − 0.999·25-s + 7.34i·29-s − 2i·31-s − 4.89·35-s + 6.92·37-s + 4.24i·41-s − 8.48·47-s + 3·49-s + ⋯ |
L(s) = 1 | − 1.09i·5-s − 0.755i·7-s + 1.47·11-s − 0.960·13-s − 1.02i·17-s − 1.58i·19-s − 1.76·23-s − 0.199·25-s + 1.36i·29-s − 0.359i·31-s − 0.828·35-s + 1.13·37-s + 0.662i·41-s − 1.23·47-s + 0.428·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.356899461\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.356899461\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.44iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 + 3.46T + 13T^{2} \) |
| 17 | \( 1 + 4.24iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 - 7.34iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - 6.92T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 8.48T + 47T^{2} \) |
| 53 | \( 1 + 2.44iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6.92T + 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 14iT - 79T^{2} \) |
| 83 | \( 1 - 4.89T + 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.854949077141330003942986048547, −7.936841272383992448601840820394, −7.10298672663370704904701167686, −6.53380308406770832396861222169, −5.34932515820125043037647120955, −4.59052447631116603015882214003, −4.07578023143905226859921117095, −2.80123508344341562061438871036, −1.47939700753215700608391284942, −0.46538809279974744719344263981,
1.70551197491143694195388001085, 2.52387674241157890923470221007, 3.68820326034178323125379235711, 4.24060984600384045894282282814, 5.72446156439710356593306045058, 6.16427458545827771772085918534, 6.86571918854837770421457514234, 7.86796194230159777077738859643, 8.396292798931557787135412008276, 9.546710624796349774219959349496