| L(s) = 1 | + 7.62·3-s − 11.9·5-s − 26.1·7-s + 31.2·9-s + 3.24·11-s − 20.0·13-s − 90.9·15-s − 17·17-s − 57.3·19-s − 199.·21-s − 77.0·23-s + 17.0·25-s + 32.1·27-s − 286.·29-s + 8.54·31-s + 24.7·33-s + 311.·35-s + 357.·37-s − 152.·39-s + 194.·41-s + 74.2·43-s − 372.·45-s − 23.6·47-s + 339.·49-s − 129.·51-s + 104.·53-s − 38.6·55-s + ⋯ |
| L(s) = 1 | + 1.46·3-s − 1.06·5-s − 1.41·7-s + 1.15·9-s + 0.0889·11-s − 0.427·13-s − 1.56·15-s − 0.242·17-s − 0.692·19-s − 2.07·21-s − 0.698·23-s + 0.136·25-s + 0.229·27-s − 1.83·29-s + 0.0495·31-s + 0.130·33-s + 1.50·35-s + 1.59·37-s − 0.628·39-s + 0.740·41-s + 0.263·43-s − 1.23·45-s − 0.0732·47-s + 0.989·49-s − 0.356·51-s + 0.270·53-s − 0.0947·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 - 7.62T + 27T^{2} \) |
| 5 | \( 1 + 11.9T + 125T^{2} \) |
| 7 | \( 1 + 26.1T + 343T^{2} \) |
| 11 | \( 1 - 3.24T + 1.33e3T^{2} \) |
| 13 | \( 1 + 20.0T + 2.19e3T^{2} \) |
| 19 | \( 1 + 57.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 77.0T + 1.21e4T^{2} \) |
| 29 | \( 1 + 286.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 8.54T + 2.97e4T^{2} \) |
| 37 | \( 1 - 357.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 194.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 74.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 23.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 104.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 249.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 370.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 939.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 520.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 348.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 953.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 486.T + 7.04e5T^{2} \) |
| 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
−10.94076462691683665047061668054, −9.654172052372214001328624595919, −9.160567057324715243015193041935, −8.011328295153391208536212759736, −7.39541081230116923429254883473, −6.16582150560964598960285167668, −4.17991056957941028672213158537, −3.47660791316925268020536526913, −2.37745099399638889191827617779, 0,
2.37745099399638889191827617779, 3.47660791316925268020536526913, 4.17991056957941028672213158537, 6.16582150560964598960285167668, 7.39541081230116923429254883473, 8.011328295153391208536212759736, 9.160567057324715243015193041935, 9.654172052372214001328624595919, 10.94076462691683665047061668054
