| L(s) = 1 | + 1.82·2-s − 3·3-s − 4.65·4-s + 5·5-s − 5.48·6-s − 7·7-s − 23.1·8-s + 9·9-s + 9.14·10-s − 64.5·11-s + 13.9·12-s − 32.3·13-s − 12.7·14-s − 15·15-s − 5.05·16-s − 56.3·17-s + 16.4·18-s − 2.74·19-s − 23.2·20-s + 21·21-s − 118.·22-s + 88.1·23-s + 69.4·24-s + 25·25-s − 59.1·26-s − 27·27-s + 32.5·28-s + ⋯ |
| L(s) = 1 | + 0.646·2-s − 0.577·3-s − 0.582·4-s + 0.447·5-s − 0.373·6-s − 0.377·7-s − 1.02·8-s + 0.333·9-s + 0.289·10-s − 1.76·11-s + 0.336·12-s − 0.690·13-s − 0.244·14-s − 0.258·15-s − 0.0790·16-s − 0.803·17-s + 0.215·18-s − 0.0331·19-s − 0.260·20-s + 0.218·21-s − 1.14·22-s + 0.799·23-s + 0.590·24-s + 0.200·25-s − 0.446·26-s − 0.192·27-s + 0.220·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 - 1.82T + 8T^{2} \) |
| 11 | \( 1 + 64.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 56.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 2.74T + 6.85e3T^{2} \) |
| 23 | \( 1 - 88.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 246.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 120.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 176.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 443.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 260.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 628.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 115.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 951.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 356.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 656.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 440.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 54.4T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 724.T + 9.12e5T^{2} \) |
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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
−13.08292458916580721099457241307, −11.98572694454539870279324252193, −10.58069027880479118600035779748, −9.718913732199968094925560978861, −8.382732867907255423730636167999, −6.78724653057891845595441544403, −5.46566543781079009701801300000, −4.70989698121031038534607877438, −2.82999919904791572750541919205, 0,
2.82999919904791572750541919205, 4.70989698121031038534607877438, 5.46566543781079009701801300000, 6.78724653057891845595441544403, 8.382732867907255423730636167999, 9.718913732199968094925560978861, 10.58069027880479118600035779748, 11.98572694454539870279324252193, 13.08292458916580721099457241307
