L(s) = 1 | − 2·3-s + 2·5-s + 9-s − 2·13-s − 4·15-s − 4·17-s + 4·19-s + 23-s − 25-s + 4·27-s + 29-s + 6·31-s + 8·37-s + 4·39-s − 6·41-s − 8·43-s + 2·45-s + 6·47-s − 7·49-s + 8·51-s + 10·53-s − 8·57-s + 4·59-s − 4·65-s − 4·67-s − 2·69-s + 12·71-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s + 1/3·9-s − 0.554·13-s − 1.03·15-s − 0.970·17-s + 0.917·19-s + 0.208·23-s − 1/5·25-s + 0.769·27-s + 0.185·29-s + 1.07·31-s + 1.31·37-s + 0.640·39-s − 0.937·41-s − 1.21·43-s + 0.298·45-s + 0.875·47-s − 49-s + 1.12·51-s + 1.37·53-s − 1.05·57-s + 0.520·59-s − 0.496·65-s − 0.488·67-s − 0.240·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.215630661\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.215630661\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−8.948620558960041218640663071576, −8.102300582113024249873544783139, −7.02786970717882749552307058094, −6.46447057301440358467069523305, −5.73620584342663723828490484157, −5.11512331768929272201616348589, −4.39512629936796575763208780173, −3.01207504624646743814496808957, −2.01965000858702629541921306620, −0.72615299814473421408066093129,
0.72615299814473421408066093129, 2.01965000858702629541921306620, 3.01207504624646743814496808957, 4.39512629936796575763208780173, 5.11512331768929272201616348589, 5.73620584342663723828490484157, 6.46447057301440358467069523305, 7.02786970717882749552307058094, 8.102300582113024249873544783139, 8.948620558960041218640663071576