| L(s) = 1 | − 1.94·2-s − 1.92·3-s − 4.21·4-s + 20.3·5-s + 3.74·6-s − 13.1·7-s + 23.7·8-s − 23.2·9-s − 39.6·10-s − 10.4·11-s + 8.10·12-s − 22.0·13-s + 25.4·14-s − 39.1·15-s − 12.5·16-s − 32.5·17-s + 45.3·18-s + 81.0·19-s − 85.8·20-s + 25.2·21-s + 20.4·22-s − 43.2·23-s − 45.7·24-s + 289.·25-s + 42.8·26-s + 96.7·27-s + 55.2·28-s + ⋯ |
| L(s) = 1 | − 0.687·2-s − 0.370·3-s − 0.526·4-s + 1.82·5-s + 0.254·6-s − 0.707·7-s + 1.05·8-s − 0.862·9-s − 1.25·10-s − 0.287·11-s + 0.195·12-s − 0.470·13-s + 0.486·14-s − 0.674·15-s − 0.195·16-s − 0.463·17-s + 0.593·18-s + 0.978·19-s − 0.959·20-s + 0.261·21-s + 0.197·22-s − 0.391·23-s − 0.388·24-s + 2.31·25-s + 0.323·26-s + 0.689·27-s + 0.372·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 1.94T + 8T^{2} \) |
| 3 | \( 1 + 1.92T + 27T^{2} \) |
| 5 | \( 1 - 20.3T + 125T^{2} \) |
| 7 | \( 1 + 13.1T + 343T^{2} \) |
| 11 | \( 1 + 10.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 32.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 81.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 43.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 263.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 269.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 228.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 23.1T + 6.89e4T^{2} \) |
| 47 | \( 1 - 143.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 438.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 446.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 680.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 91.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.02e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 342.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 816.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.11e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 632.T + 9.12e5T^{2} \) |
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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
−8.825953804510937473004741851537, −7.84937346140893723954684180494, −6.79775708663133134126620998195, −6.03730862967466596237057116308, −5.38655891834541798683400112652, −4.68037105832242891019631735409, −3.13898376053568923739654259722, −2.25062727960313333665546363277, −1.10324729745582311485164354926, 0,
1.10324729745582311485164354926, 2.25062727960313333665546363277, 3.13898376053568923739654259722, 4.68037105832242891019631735409, 5.38655891834541798683400112652, 6.03730862967466596237057116308, 6.79775708663133134126620998195, 7.84937346140893723954684180494, 8.825953804510937473004741851537
