L(s) = 1 | − 2-s − 2·3-s + 4-s − 3·5-s + 2·6-s − 4·7-s − 8-s + 9-s + 3·10-s − 2·11-s − 2·12-s − 13-s + 4·14-s + 6·15-s + 16-s − 6·17-s − 18-s − 4·19-s − 3·20-s + 8·21-s + 2·22-s − 9·23-s + 2·24-s + 4·25-s + 26-s + 4·27-s − 4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s − 1.34·5-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.948·10-s − 0.603·11-s − 0.577·12-s − 0.277·13-s + 1.06·14-s + 1.54·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.670·20-s + 1.74·21-s + 0.426·22-s − 1.87·23-s + 0.408·24-s + 4/5·25-s + 0.196·26-s + 0.769·27-s − 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31478 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31478 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 15739 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 11 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.96157834516836, −15.69435472291278, −15.08103436267391, −14.52630891697278, −13.55301720842949, −12.85627930498837, −12.67684776592718, −12.03522575440884, −11.49584059895362, −11.15244581201941, −10.54823120832840, −10.12938795028159, −9.496885177769871, −8.765728940046544, −8.363966065104031, −7.604326513668194, −7.069084585578214, −6.557613375643603, −6.125970678169867, −5.408232948361362, −4.719651867365276, −3.757255540084210, −3.598924365508598, −2.504043641997019, −1.773887930464119, 0, 0, 0,
1.773887930464119, 2.504043641997019, 3.598924365508598, 3.757255540084210, 4.719651867365276, 5.408232948361362, 6.125970678169867, 6.557613375643603, 7.069084585578214, 7.604326513668194, 8.363966065104031, 8.765728940046544, 9.496885177769871, 10.12938795028159, 10.54823120832840, 11.15244581201941, 11.49584059895362, 12.03522575440884, 12.67684776592718, 12.85627930498837, 13.55301720842949, 14.52630891697278, 15.08103436267391, 15.69435472291278, 15.96157834516836