L(s) = 1 | + 2-s + 3-s − 4-s + 6-s + 3·7-s − 3·8-s − 2·9-s + 3·11-s − 12-s + 6·13-s + 3·14-s − 16-s − 5·17-s − 2·18-s + 2·19-s + 3·21-s + 3·22-s + 4·23-s − 3·24-s + 6·26-s − 5·27-s − 3·28-s − 7·29-s + 5·31-s + 5·32-s + 3·33-s − 5·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 1.13·7-s − 1.06·8-s − 2/3·9-s + 0.904·11-s − 0.288·12-s + 1.66·13-s + 0.801·14-s − 1/4·16-s − 1.21·17-s − 0.471·18-s + 0.458·19-s + 0.654·21-s + 0.639·22-s + 0.834·23-s − 0.612·24-s + 1.17·26-s − 0.962·27-s − 0.566·28-s − 1.29·29-s + 0.898·31-s + 0.883·32-s + 0.522·33-s − 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.018216739\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.018216739\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 83 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.017571364405704341140281754032, −8.453475283691595638097668301696, −7.80720406011570304437105683036, −6.51003986895949489621827706154, −5.87263406714637014971627456813, −4.97148776811947809940547323320, −4.13105076781702502762968049773, −3.53133508802941266584945160071, −2.41615676868067838581180557878, −1.09189632316430757900963074575,
1.09189632316430757900963074575, 2.41615676868067838581180557878, 3.53133508802941266584945160071, 4.13105076781702502762968049773, 4.97148776811947809940547323320, 5.87263406714637014971627456813, 6.51003986895949489621827706154, 7.80720406011570304437105683036, 8.453475283691595638097668301696, 9.017571364405704341140281754032