| L(s) = 1 | + 1.71·2-s + 3·3-s − 5.05·4-s − 8.99·5-s + 5.15·6-s + 15.9·7-s − 22.4·8-s + 9·9-s − 15.4·10-s − 11·11-s − 15.1·12-s − 55.9·13-s + 27.3·14-s − 26.9·15-s + 1.90·16-s + 66.4·17-s + 15.4·18-s + 106.·19-s + 45.4·20-s + 47.8·21-s − 18.8·22-s + 139.·23-s − 67.2·24-s − 44.0·25-s − 96.1·26-s + 27·27-s − 80.5·28-s + ⋯ |
| L(s) = 1 | + 0.607·2-s + 0.577·3-s − 0.631·4-s − 0.804·5-s + 0.350·6-s + 0.860·7-s − 0.990·8-s + 0.333·9-s − 0.488·10-s − 0.301·11-s − 0.364·12-s − 1.19·13-s + 0.522·14-s − 0.464·15-s + 0.0297·16-s + 0.948·17-s + 0.202·18-s + 1.28·19-s + 0.508·20-s + 0.496·21-s − 0.183·22-s + 1.26·23-s − 0.571·24-s − 0.352·25-s − 0.725·26-s + 0.192·27-s − 0.543·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 1.71T + 8T^{2} \) |
| 5 | \( 1 + 8.99T + 125T^{2} \) |
| 7 | \( 1 - 15.9T + 343T^{2} \) |
| 13 | \( 1 + 55.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 66.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 106.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 181.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.5T + 2.97e4T^{2} \) |
| 37 | \( 1 - 82.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 367.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 263.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 266.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 13.6T + 1.48e5T^{2} \) |
| 59 | \( 1 - 446.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 935.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 317.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 337.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 126.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 339.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 337.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.04e3T + 9.12e5T^{2} \) |
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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
−8.190891360832076809841934336350, −7.73865402007195826767989525663, −7.12426382901583580344731664384, −5.60348457937127974547518727966, −5.06585205703552631056896224005, −4.32744777898155813186064365364, −3.45268344413865000240111941304, −2.73292530040019778413807153093, −1.27751576758087861573432726321, 0,
1.27751576758087861573432726321, 2.73292530040019778413807153093, 3.45268344413865000240111941304, 4.32744777898155813186064365364, 5.06585205703552631056896224005, 5.60348457937127974547518727966, 7.12426382901583580344731664384, 7.73865402007195826767989525663, 8.190891360832076809841934336350
