| L(s) = 1 | − 5.15·2-s + 6.49·3-s + 18.5·4-s − 17.2·5-s − 33.4·6-s + 23.3·7-s − 54.3·8-s + 15.2·9-s + 88.9·10-s − 60.5·11-s + 120.·12-s + 10.9·13-s − 120.·14-s − 112.·15-s + 131.·16-s − 3.57·17-s − 78.5·18-s − 33.2·19-s − 320.·20-s + 151.·21-s + 312.·22-s + 63.7·23-s − 353.·24-s + 173.·25-s − 56.3·26-s − 76.4·27-s + 432.·28-s + ⋯ |
| L(s) = 1 | − 1.82·2-s + 1.25·3-s + 2.31·4-s − 1.54·5-s − 2.27·6-s + 1.25·7-s − 2.40·8-s + 0.564·9-s + 2.81·10-s − 1.65·11-s + 2.90·12-s + 0.233·13-s − 2.29·14-s − 1.93·15-s + 2.05·16-s − 0.0509·17-s − 1.02·18-s − 0.401·19-s − 3.58·20-s + 1.57·21-s + 3.02·22-s + 0.577·23-s − 3.00·24-s + 1.38·25-s − 0.425·26-s − 0.544·27-s + 2.91·28-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 | \( 1 + 5.15T + 8T^{2} \) |
| 3 | \( 1 - 6.49T + 27T^{2} \) |
| 5 | \( 1 + 17.2T + 125T^{2} \) |
| 7 | \( 1 - 23.3T + 343T^{2} \) |
| 11 | \( 1 + 60.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 3.57T + 4.91e3T^{2} \) |
| 19 | \( 1 + 33.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 63.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 89.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 222.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 59.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 143.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 379.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 150.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 207.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 486.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 13.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 411.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 813.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 350.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
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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
−8.457030567060497136698090483483, −7.86464794465640330515895008165, −7.70453692642155580680155821497, −6.76543799566213383274270348205, −5.19068614911375227805954277369, −4.10276491202840932457324528831, −2.90231362207427521872075808695, −2.33951493916130804209633802911, −1.08849496375378631339522059928, 0,
1.08849496375378631339522059928, 2.33951493916130804209633802911, 2.90231362207427521872075808695, 4.10276491202840932457324528831, 5.19068614911375227805954277369, 6.76543799566213383274270348205, 7.70453692642155580680155821497, 7.86464794465640330515895008165, 8.457030567060497136698090483483
