| L(s) = 1 | + 3.52·2-s + 3.06·3-s + 4.44·4-s + 14.8·5-s + 10.8·6-s + 18.1·7-s − 12.5·8-s − 17.6·9-s + 52.4·10-s − 54.4·11-s + 13.6·12-s − 71.7·13-s + 64.0·14-s + 45.5·15-s − 79.7·16-s − 70.3·17-s − 62.1·18-s + 71.5·19-s + 66.1·20-s + 55.6·21-s − 192.·22-s + 107.·23-s − 38.3·24-s + 95.6·25-s − 253.·26-s − 136.·27-s + 80.8·28-s + ⋯ |
| L(s) = 1 | + 1.24·2-s + 0.589·3-s + 0.556·4-s + 1.32·5-s + 0.735·6-s + 0.980·7-s − 0.553·8-s − 0.652·9-s + 1.65·10-s − 1.49·11-s + 0.327·12-s − 1.53·13-s + 1.22·14-s + 0.783·15-s − 1.24·16-s − 1.00·17-s − 0.813·18-s + 0.863·19-s + 0.739·20-s + 0.578·21-s − 1.86·22-s + 0.977·23-s − 0.326·24-s + 0.765·25-s − 1.90·26-s − 0.974·27-s + 0.545·28-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 | \( 1 - 3.52T + 8T^{2} \) |
| 3 | \( 1 - 3.06T + 27T^{2} \) |
| 5 | \( 1 - 14.8T + 125T^{2} \) |
| 7 | \( 1 - 18.1T + 343T^{2} \) |
| 11 | \( 1 + 54.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 71.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 70.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 165.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 24.0T + 6.89e4T^{2} \) |
| 47 | \( 1 + 454.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 662.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 325.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 545.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 976.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 494.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 462.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 51.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 391.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 685.T + 9.12e5T^{2} \) |
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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
−8.472562579477488233188175341345, −7.66469482448547715687227675722, −6.75245747148580527464089180055, −5.55509175209834397011960883511, −5.25981722978559229775515243101, −4.69480524625456538073362660733, −3.27574402216578242397042159679, −2.47586822512957221662005552450, −2.01434140289342515539777310071, 0,
2.01434140289342515539777310071, 2.47586822512957221662005552450, 3.27574402216578242397042159679, 4.69480524625456538073362660733, 5.25981722978559229775515243101, 5.55509175209834397011960883511, 6.75245747148580527464089180055, 7.66469482448547715687227675722, 8.472562579477488233188175341345
