L(s) = 1 | + 2·2-s + 3-s + 2·4-s + 5-s + 2·6-s − 7-s + 9-s + 2·10-s − 2·11-s + 2·12-s − 13-s − 2·14-s + 15-s − 4·16-s + 2·18-s + 19-s + 2·20-s − 21-s − 4·22-s + 3·23-s − 4·25-s − 2·26-s + 27-s − 2·28-s − 5·29-s + 2·30-s + 9·31-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s + 0.632·10-s − 0.603·11-s + 0.577·12-s − 0.277·13-s − 0.534·14-s + 0.258·15-s − 16-s + 0.471·18-s + 0.229·19-s + 0.447·20-s − 0.218·21-s − 0.852·22-s + 0.625·23-s − 4/5·25-s − 0.392·26-s + 0.192·27-s − 0.377·28-s − 0.928·29-s + 0.365·30-s + 1.61·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.852768318\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.852768318\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 17 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−12.28936108770676961423612040714, −11.22516944735448501879908089914, −9.997612005245311033735868727795, −9.153614989507917020414880699166, −7.85096152981510062691288793912, −6.66540549198067832227195282850, −5.65939043191698227214275856071, −4.64274960483946295855931167095, −3.41858543863816620367865257858, −2.38895973908794646217186104367,
2.38895973908794646217186104367, 3.41858543863816620367865257858, 4.64274960483946295855931167095, 5.65939043191698227214275856071, 6.66540549198067832227195282850, 7.85096152981510062691288793912, 9.153614989507917020414880699166, 9.997612005245311033735868727795, 11.22516944735448501879908089914, 12.28936108770676961423612040714