L(s) = 1 | + 3.37·2-s + 3.38·4-s − 13.8·5-s + 24.5·7-s − 15.5·8-s − 46.7·10-s − 22.1·11-s − 22.1·13-s + 82.8·14-s − 79.6·16-s − 91.9·17-s − 37.0·19-s − 46.8·20-s − 74.7·22-s + 143.·23-s + 66.9·25-s − 74.6·26-s + 83.0·28-s + 196.·29-s − 216.·31-s − 143.·32-s − 310.·34-s − 340.·35-s + 147.·37-s − 125.·38-s + 215.·40-s + 220.·41-s + ⋯ |
L(s) = 1 | + 1.19·2-s + 0.422·4-s − 1.23·5-s + 1.32·7-s − 0.688·8-s − 1.47·10-s − 0.607·11-s − 0.472·13-s + 1.58·14-s − 1.24·16-s − 1.31·17-s − 0.447·19-s − 0.523·20-s − 0.724·22-s + 1.30·23-s + 0.535·25-s − 0.563·26-s + 0.560·28-s + 1.25·29-s − 1.25·31-s − 0.795·32-s − 1.56·34-s − 1.64·35-s + 0.655·37-s − 0.533·38-s + 0.853·40-s + 0.840·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.414857424\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.414857424\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
good | 2 | \( 1 - 3.37T + 8T^{2} \) |
| 5 | \( 1 + 13.8T + 125T^{2} \) |
| 7 | \( 1 - 24.5T + 343T^{2} \) |
| 11 | \( 1 + 22.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 91.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 37.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 143.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 216.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 147.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 220.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 357.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 250.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 606.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 104.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 57.7T + 3.00e5T^{2} \) |
| 71 | \( 1 - 51.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 471.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 876.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 178.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 63.1T + 9.12e5T^{2} \) |
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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
−8.641877948701231453582256093926, −7.80213374556086254317865392976, −7.22702456308796000620578311652, −6.21017063298471062324102965178, −5.16322076886907515722845983266, −4.58291331189627092249037523120, −4.18296118442003817146721982597, −3.05822156522732798967402048465, −2.18239005665922758135807193859, −0.56796434642962570911127629614,
0.56796434642962570911127629614, 2.18239005665922758135807193859, 3.05822156522732798967402048465, 4.18296118442003817146721982597, 4.58291331189627092249037523120, 5.16322076886907515722845983266, 6.21017063298471062324102965178, 7.22702456308796000620578311652, 7.80213374556086254317865392976, 8.641877948701231453582256093926