L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s + 3·7-s − 8-s + 9-s + 4·10-s + 5·11-s − 12-s + 3·13-s − 3·14-s + 4·15-s + 16-s + 3·17-s − 18-s − 7·19-s − 4·20-s − 3·21-s − 5·22-s + 9·23-s + 24-s + 11·25-s − 3·26-s − 27-s + 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 1.50·11-s − 0.288·12-s + 0.832·13-s − 0.801·14-s + 1.03·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.60·19-s − 0.894·20-s − 0.654·21-s − 1.06·22-s + 1.87·23-s + 0.204·24-s + 11/5·25-s − 0.588·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6807637972\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6807637972\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−11.88212339495344719301034550255, −11.19169781681082454956006126222, −10.83779638925013083318167734345, −9.046947615550729857916202382850, −8.318914495123520315268044799592, −7.40369435085475529586066660482, −6.42380409964038574403615594445, −4.69516264772634405387664054544, −3.69413280585934281517510849684, −1.13658449571290171220917545956,
1.13658449571290171220917545956, 3.69413280585934281517510849684, 4.69516264772634405387664054544, 6.42380409964038574403615594445, 7.40369435085475529586066660482, 8.318914495123520315268044799592, 9.046947615550729857916202382850, 10.83779638925013083318167734345, 11.19169781681082454956006126222, 11.88212339495344719301034550255