L(s) = 1 | + 1.46·2-s − 5.85·4-s − 9.85·5-s + 29.9·7-s − 20.3·8-s − 14.4·10-s − 46.9·11-s + 43.8·14-s + 17.0·16-s + 48.2·17-s + 120.·19-s + 57.6·20-s − 68.7·22-s − 130.·23-s − 27.9·25-s − 175.·28-s + 194.·29-s − 32.0·31-s + 187.·32-s + 70.7·34-s − 294.·35-s − 32.4·37-s + 176.·38-s + 200.·40-s + 241.·41-s + 96.4·43-s + 274.·44-s + ⋯ |
L(s) = 1 | + 0.518·2-s − 0.731·4-s − 0.881·5-s + 1.61·7-s − 0.897·8-s − 0.456·10-s − 1.28·11-s + 0.837·14-s + 0.266·16-s + 0.688·17-s + 1.45·19-s + 0.644·20-s − 0.666·22-s − 1.18·23-s − 0.223·25-s − 1.18·28-s + 1.24·29-s − 0.185·31-s + 1.03·32-s + 0.356·34-s − 1.42·35-s − 0.144·37-s + 0.752·38-s + 0.790·40-s + 0.921·41-s + 0.341·43-s + 0.940·44-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
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.46T + 8T^{2} \) |
| 5 | \( 1 + 9.85T + 125T^{2} \) |
| 7 | \( 1 - 29.9T + 343T^{2} \) |
| 11 | \( 1 + 46.9T + 1.33e3T^{2} \) |
| 17 | \( 1 - 48.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 120.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 194.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 32.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 32.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 241.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 96.4T + 7.95e4T^{2} \) |
| 47 | \( 1 + 539.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 152.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 327.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 98.4T + 2.26e5T^{2} \) |
| 67 | \( 1 + 441.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 345.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 773.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 150.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 337.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 169.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 214.T + 9.12e5T^{2} \) |
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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
−8.309029116802398372080305134743, −8.021136345663474963516144953087, −7.44347447243537547688743538088, −5.86996071652431033583379712785, −5.15345175896201926381732051056, −4.60797453178107970733814881625, −3.72903590617930109952302730860, −2.71773770013627145978133151951, −1.23411088150481203830956589587, 0,
1.23411088150481203830956589587, 2.71773770013627145978133151951, 3.72903590617930109952302730860, 4.60797453178107970733814881625, 5.15345175896201926381732051056, 5.86996071652431033583379712785, 7.44347447243537547688743538088, 8.021136345663474963516144953087, 8.309029116802398372080305134743