| L(s) = 1 | + 1.42·2-s − 6.49·3-s − 5.95·4-s − 0.801·5-s − 9.29·6-s − 16.7·7-s − 19.9·8-s + 15.2·9-s − 1.14·10-s − 11·11-s + 38.7·12-s − 23.8·14-s + 5.20·15-s + 19.1·16-s + 43.7·17-s + 21.7·18-s − 10.5·19-s + 4.77·20-s + 108.·21-s − 15.7·22-s − 157.·23-s + 129.·24-s − 124.·25-s + 76.4·27-s + 99.5·28-s + 52.1·29-s + 7.44·30-s + ⋯ |
| L(s) = 1 | + 0.505·2-s − 1.25·3-s − 0.744·4-s − 0.0716·5-s − 0.632·6-s − 0.902·7-s − 0.881·8-s + 0.564·9-s − 0.0362·10-s − 0.301·11-s + 0.931·12-s − 0.456·14-s + 0.0896·15-s + 0.298·16-s + 0.623·17-s + 0.285·18-s − 0.126·19-s + 0.0533·20-s + 1.12·21-s − 0.152·22-s − 1.42·23-s + 1.10·24-s − 0.994·25-s + 0.544·27-s + 0.671·28-s + 0.333·29-s + 0.0453·30-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 | \( 1 - 1.42T + 8T^{2} \) |
| 3 | \( 1 + 6.49T + 27T^{2} \) |
| 5 | \( 1 + 0.801T + 125T^{2} \) |
| 7 | \( 1 + 16.7T + 343T^{2} \) |
| 17 | \( 1 - 43.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 10.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.1T + 2.43e4T^{2} \) |
| 31 | \( 1 - 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 23.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 59.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 337.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 364.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 493.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 414.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 322.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 39.5T + 3.00e5T^{2} \) |
| 71 | \( 1 + 319.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 213.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 786.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 677.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 336.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−8.475642192227705430586766698506, −7.64667542173137261356453923946, −6.46237876410768564067244334616, −5.94813487347406520610563143494, −5.38701385539849058813520562549, −4.42416118093241765052778362055, −3.71460607376602964624027260945, −2.60409466679775919375120327431, −0.816608158004391469116347576264, 0,
0.816608158004391469116347576264, 2.60409466679775919375120327431, 3.71460607376602964624027260945, 4.42416118093241765052778362055, 5.38701385539849058813520562549, 5.94813487347406520610563143494, 6.46237876410768564067244334616, 7.64667542173137261356453923946, 8.475642192227705430586766698506
