| L(s) = 1 | + i·5-s − 1.73i·7-s + 3.46·11-s + 13-s + 1.73i·19-s + 3.46·23-s − 25-s + 6i·29-s − 3.46i·31-s + 1.73·35-s − 7·37-s − 6i·41-s − 3.46i·43-s + 6.92·47-s + 4·49-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s − 0.654i·7-s + 1.04·11-s + 0.277·13-s + 0.397i·19-s + 0.722·23-s − 0.200·25-s + 1.11i·29-s − 0.622i·31-s + 0.292·35-s − 1.15·37-s − 0.937i·41-s − 0.528i·43-s + 1.01·47-s + 0.571·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.925927655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.925927655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| good | 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 5.19iT - 67T^{2} \) |
| 71 | \( 1 - 3.46T + 71T^{2} \) |
| 73 | \( 1 - T + 73T^{2} \) |
| 79 | \( 1 - 8.66iT - 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - T + 97T^{2} \) |
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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
−9.003144846143837835516842773062, −8.423850186662513811207976314202, −7.20021097924390878549922567799, −6.98737462720311130208889987448, −5.99532774583619869436576379465, −5.10411608875379806072954311305, −3.94408378957739154941273383026, −3.49633787112694568590308516523, −2.14178722861798386979094389402, −0.937952479543584697810901185521,
0.963429626694168764341601162701, 2.13694360238549647192676374937, 3.27462571662969934193773560785, 4.24107494207845485717800315905, 5.07804056398547159204739192426, 5.94745949388314923772472107160, 6.65748250562851938265639058011, 7.53674134696136161783049568137, 8.550581290872616602591855266784, 8.970392105299087083794673635712
