| L(s) = 1 | + 3-s + 7-s + 9-s − 6·11-s + 5·13-s + 6·17-s − 5·19-s + 21-s + 6·23-s + 27-s + 6·29-s − 31-s − 6·33-s + 2·37-s + 5·39-s − 43-s − 6·47-s − 6·49-s + 6·51-s − 12·53-s − 5·57-s + 6·59-s + 13·61-s + 63-s + 11·67-s + 6·69-s − 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.38·13-s + 1.45·17-s − 1.14·19-s + 0.218·21-s + 1.25·23-s + 0.192·27-s + 1.11·29-s − 0.179·31-s − 1.04·33-s + 0.328·37-s + 0.800·39-s − 0.152·43-s − 0.875·47-s − 6/7·49-s + 0.840·51-s − 1.64·53-s − 0.662·57-s + 0.781·59-s + 1.66·61-s + 0.125·63-s + 1.34·67-s + 0.722·69-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.573961435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.573961435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.217074462333978304679318066155, −7.86031613752285574110900894478, −6.90853508274228096394873147119, −6.10240622509958663752088302808, −5.24366870821784549077339631128, −4.67219405629838718258752085064, −3.53493989661310366851205708196, −2.96682092923381496870383001501, −1.99382224727202961936044120648, −0.881301861460005392759052490677,
0.881301861460005392759052490677, 1.99382224727202961936044120648, 2.96682092923381496870383001501, 3.53493989661310366851205708196, 4.67219405629838718258752085064, 5.24366870821784549077339631128, 6.10240622509958663752088302808, 6.90853508274228096394873147119, 7.86031613752285574110900894478, 8.217074462333978304679318066155
