L(s) = 1 | − 4.12·2-s + 9·4-s + 3.05·5-s − 6.68·7-s − 4.12·8-s − 12.5·10-s − 32.2·11-s + 27.5·14-s − 55·16-s + 28.8·17-s − 101.·19-s + 27.4·20-s + 132.·22-s + 118.·23-s − 115.·25-s − 60.1·28-s − 160.·29-s − 38.0·31-s + 259.·32-s − 118.·34-s − 20.3·35-s − 327.·37-s + 417.·38-s − 12.5·40-s − 56.0·41-s + 127.·43-s − 290.·44-s + ⋯ |
L(s) = 1 | − 1.45·2-s + 1.12·4-s + 0.272·5-s − 0.360·7-s − 0.182·8-s − 0.397·10-s − 0.883·11-s + 0.526·14-s − 0.859·16-s + 0.411·17-s − 1.22·19-s + 0.306·20-s + 1.28·22-s + 1.07·23-s − 0.925·25-s − 0.405·28-s − 1.02·29-s − 0.220·31-s + 1.43·32-s − 0.600·34-s − 0.0984·35-s − 1.45·37-s + 1.78·38-s − 0.0497·40-s − 0.213·41-s + 0.453·43-s − 0.994·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.5318408528\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5318408528\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 4.12T + 8T^{2} \) |
| 5 | \( 1 - 3.05T + 125T^{2} \) |
| 7 | \( 1 + 6.68T + 343T^{2} \) |
| 11 | \( 1 + 32.2T + 1.33e3T^{2} \) |
| 17 | \( 1 - 28.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 118.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 160.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 38.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 56.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 517.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 695.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 656.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 701.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 57.1T + 3.00e5T^{2} \) |
| 71 | \( 1 + 309.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 389.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 901.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 687.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.75e3T + 9.12e5T^{2} \) |
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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
−9.030149940915905097031796152418, −8.518471152298397939358354235427, −7.58300543859358617563331709750, −7.05675908518743857300184197537, −6.01937795329651416388540407111, −5.11461393272137153443034943744, −3.88254955175204368082873557903, −2.59231736481393849662359922981, −1.72010638527191251343467174974, −0.43696791310155239604860135844,
0.43696791310155239604860135844, 1.72010638527191251343467174974, 2.59231736481393849662359922981, 3.88254955175204368082873557903, 5.11461393272137153443034943744, 6.01937795329651416388540407111, 7.05675908518743857300184197537, 7.58300543859358617563331709750, 8.518471152298397939358354235427, 9.030149940915905097031796152418