L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 3·7-s − 8-s − 2·9-s + 10-s + 4·11-s − 12-s − 7·13-s + 3·14-s + 15-s + 16-s − 4·17-s + 2·18-s − 6·19-s − 20-s + 3·21-s − 4·22-s + 6·23-s + 24-s − 4·25-s + 7·26-s + 5·27-s − 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s − 1.94·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.471·18-s − 1.37·19-s − 0.223·20-s + 0.654·21-s − 0.852·22-s + 1.25·23-s + 0.204·24-s − 4/5·25-s + 1.37·26-s + 0.962·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 158 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 79 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 11 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−12.05634195288322895320916696017, −11.54480511636342312793408195564, −10.28684063846692777156990815161, −9.410151387289790151865050532885, −8.410553722468251933217974583242, −6.89817659603665535966423256063, −6.33379138631449939041301566310, −4.58010875049676022597033523034, −2.77554204201998599512986390174, 0,
2.77554204201998599512986390174, 4.58010875049676022597033523034, 6.33379138631449939041301566310, 6.89817659603665535966423256063, 8.410553722468251933217974583242, 9.410151387289790151865050532885, 10.28684063846692777156990815161, 11.54480511636342312793408195564, 12.05634195288322895320916696017