| L(s) = 1 | + 2.34·2-s − 6.88·3-s − 2.49·4-s − 4.99·5-s − 16.1·6-s − 27.5·7-s − 24.6·8-s + 20.4·9-s − 11.7·10-s − 11·11-s + 17.1·12-s − 64.7·14-s + 34.4·15-s − 37.8·16-s + 109.·17-s + 47.9·18-s − 31.0·19-s + 12.4·20-s + 190.·21-s − 25.8·22-s + 125.·23-s + 169.·24-s − 100.·25-s + 45.3·27-s + 68.7·28-s − 123.·29-s + 80.7·30-s + ⋯ |
| L(s) = 1 | + 0.829·2-s − 1.32·3-s − 0.311·4-s − 0.447·5-s − 1.09·6-s − 1.48·7-s − 1.08·8-s + 0.756·9-s − 0.371·10-s − 0.301·11-s + 0.412·12-s − 1.23·14-s + 0.592·15-s − 0.591·16-s + 1.56·17-s + 0.627·18-s − 0.375·19-s + 0.139·20-s + 1.97·21-s − 0.250·22-s + 1.13·23-s + 1.44·24-s − 0.800·25-s + 0.323·27-s + 0.463·28-s − 0.789·29-s + 0.491·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.34T + 8T^{2} \) |
| 3 | \( 1 + 6.88T + 27T^{2} \) |
| 5 | \( 1 + 4.99T + 125T^{2} \) |
| 7 | \( 1 + 27.5T + 343T^{2} \) |
| 17 | \( 1 - 109.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 31.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 123.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 69.3T + 2.97e4T^{2} \) |
| 37 | \( 1 - 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 140.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 244.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 456.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 764.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 589.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 577.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 403.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 235.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 568.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 143.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 331.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 154.T + 9.12e5T^{2} \) |
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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
−8.501079498977879851643875632860, −7.40912178834588628255572445032, −6.56112950686727114163869229852, −5.82927374066988929602501423778, −5.42984659707959640179427811972, −4.44363973618207756431294092018, −3.57967176255623726852615751905, −2.86784278863917710124220075200, −0.802970458010110584719445361370, 0,
0.802970458010110584719445361370, 2.86784278863917710124220075200, 3.57967176255623726852615751905, 4.44363973618207756431294092018, 5.42984659707959640179427811972, 5.82927374066988929602501423778, 6.56112950686727114163869229852, 7.40912178834588628255572445032, 8.501079498977879851643875632860
