| L(s) = 1 | − 2-s − 3-s − 4-s + 2·5-s + 6-s + 3·8-s + 9-s − 2·10-s + 4·11-s + 12-s + 2·13-s − 2·15-s − 16-s + 6·17-s − 18-s − 4·19-s − 2·20-s − 4·22-s − 3·24-s − 25-s − 2·26-s − 27-s − 2·29-s + 2·30-s − 5·32-s − 4·33-s − 6·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 0.516·15-s − 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.447·20-s − 0.852·22-s − 0.612·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.371·29-s + 0.365·30-s − 0.883·32-s − 0.696·33-s − 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7222862454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7222862454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.09697998821175856125894317781, −11.98659319295797721290412051260, −10.77789244508403227045944829665, −9.863214598051417092763914791206, −9.163045893798789171919999202805, −7.977686193886306753143677629185, −6.53484921474790582970192984243, −5.48510765590071063887274763802, −4.01671863219499060051377207353, −1.40786771277750203137263322761,
1.40786771277750203137263322761, 4.01671863219499060051377207353, 5.48510765590071063887274763802, 6.53484921474790582970192984243, 7.977686193886306753143677629185, 9.163045893798789171919999202805, 9.863214598051417092763914791206, 10.77789244508403227045944829665, 11.98659319295797721290412051260, 13.09697998821175856125894317781
