L(s) = 1 | − 1.00·2-s − 2.78·3-s − 0.983·4-s − 0.698·5-s + 2.80·6-s + 4.38·7-s + 3.00·8-s + 4.75·9-s + 0.704·10-s + 2.73·12-s + 5.97·13-s − 4.42·14-s + 1.94·15-s − 1.06·16-s + 3.19·17-s − 4.79·18-s − 2.44·19-s + 0.687·20-s − 12.2·21-s − 6.45·23-s − 8.37·24-s − 4.51·25-s − 6.02·26-s − 4.87·27-s − 4.31·28-s + 3.14·29-s − 1.96·30-s + ⋯ |
L(s) = 1 | − 0.713·2-s − 1.60·3-s − 0.491·4-s − 0.312·5-s + 1.14·6-s + 1.65·7-s + 1.06·8-s + 1.58·9-s + 0.222·10-s + 0.790·12-s + 1.65·13-s − 1.18·14-s + 0.502·15-s − 0.266·16-s + 0.775·17-s − 1.12·18-s − 0.561·19-s + 0.153·20-s − 2.66·21-s − 1.34·23-s − 1.70·24-s − 0.902·25-s − 1.18·26-s − 0.939·27-s − 0.815·28-s + 0.583·29-s − 0.358·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6731545242\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6731545242\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 + 1.00T + 2T^{2} \) |
| 3 | \( 1 + 2.78T + 3T^{2} \) |
| 5 | \( 1 + 0.698T + 5T^{2} \) |
| 7 | \( 1 - 4.38T + 7T^{2} \) |
| 13 | \( 1 - 5.97T + 13T^{2} \) |
| 17 | \( 1 - 3.19T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 6.45T + 23T^{2} \) |
| 29 | \( 1 - 3.14T + 29T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 - 6.29T + 37T^{2} \) |
| 41 | \( 1 + 5.15T + 41T^{2} \) |
| 43 | \( 1 - 9.20T + 43T^{2} \) |
| 47 | \( 1 - 3.64T + 47T^{2} \) |
| 53 | \( 1 - 9.56T + 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 + 1.51T + 61T^{2} \) |
| 67 | \( 1 + 8.58T + 67T^{2} \) |
| 71 | \( 1 + 7.32T + 71T^{2} \) |
| 73 | \( 1 + 1.53T + 73T^{2} \) |
| 79 | \( 1 + 1.04T + 79T^{2} \) |
| 83 | \( 1 - 7.02T + 83T^{2} \) |
| 89 | \( 1 - 0.863T + 89T^{2} \) |
| 97 | \( 1 + 7.02T + 97T^{2} \) |
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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
−9.818024973212292851949741929610, −8.598877368514850547040466061646, −8.115079740450366571093137975767, −7.38800244118204543838932257111, −6.07565289931806450496126904478, −5.57787193330941072245125067083, −4.48936669874864895159473024089, −4.08520732939388283197573739192, −1.64670824421502845529333165509, −0.801105987084971100371129979929,
0.801105987084971100371129979929, 1.64670824421502845529333165509, 4.08520732939388283197573739192, 4.48936669874864895159473024089, 5.57787193330941072245125067083, 6.07565289931806450496126904478, 7.38800244118204543838932257111, 8.115079740450366571093137975767, 8.598877368514850547040466061646, 9.818024973212292851949741929610