L(s) = 1 | − 4.12·2-s − 3·3-s + 9·4-s + 3.05·5-s + 12.3·6-s + 6.68·7-s − 4.12·8-s + 9·9-s − 12.5·10-s − 32.2·11-s − 27·12-s − 27.5·14-s − 9.15·15-s − 55·16-s − 28.8·17-s − 37.1·18-s + 101.·19-s + 27.4·20-s − 20.0·21-s + 132.·22-s − 118.·23-s + 12.3·24-s − 115.·25-s − 27·27-s + 60.1·28-s + 160.·29-s + 37.7·30-s + ⋯ |
L(s) = 1 | − 1.45·2-s − 0.577·3-s + 1.12·4-s + 0.272·5-s + 0.841·6-s + 0.360·7-s − 0.182·8-s + 0.333·9-s − 0.397·10-s − 0.883·11-s − 0.649·12-s − 0.526·14-s − 0.157·15-s − 0.859·16-s − 0.411·17-s − 0.485·18-s + 1.22·19-s + 0.306·20-s − 0.208·21-s + 1.28·22-s − 1.07·23-s + 0.105·24-s − 0.925·25-s − 0.192·27-s + 0.405·28-s + 1.02·29-s + 0.229·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 3T \) |
| 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 | \( 1 + 1.75e3T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.921322975442056107758758306482, −9.398780049865940325881810726404, −8.116231014237500435199712366652, −7.71623025900656791641355529437, −6.56719344942428786780915256712, −5.52317212175414376291587927734, −4.39872582696397137206031941712, −2.51229610515168250121098752382, −1.26249770485836946364718737188, 0,
1.26249770485836946364718737188, 2.51229610515168250121098752382, 4.39872582696397137206031941712, 5.52317212175414376291587927734, 6.56719344942428786780915256712, 7.71623025900656791641355529437, 8.116231014237500435199712366652, 9.398780049865940325881810726404, 9.921322975442056107758758306482