| L(s) = 1 | − 3-s − 4·7-s + 9-s + 4·11-s + 13-s − 2·17-s − 4·19-s + 4·21-s − 27-s + 6·29-s − 4·33-s − 10·37-s − 39-s + 2·41-s + 4·43-s − 4·47-s + 9·49-s + 2·51-s + 6·53-s + 4·57-s − 4·59-s − 2·61-s − 4·63-s + 4·67-s − 8·71-s + 2·73-s − 16·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.485·17-s − 0.917·19-s + 0.872·21-s − 0.192·27-s + 1.11·29-s − 0.696·33-s − 1.64·37-s − 0.160·39-s + 0.312·41-s + 0.609·43-s − 0.583·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.256·61-s − 0.503·63-s + 0.488·67-s − 0.949·71-s + 0.234·73-s − 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.45746563052728, −14.95555110349994, −14.18365637101948, −13.74156662388855, −13.17251665501460, −12.61370143947619, −12.22218396546880, −11.76022117635415, −11.04586125940362, −10.48979757200102, −10.08716942994901, −9.384660244495443, −8.966798364760700, −8.475853762791593, −7.552272640552805, −6.777888900560858, −6.530357370982660, −6.162577587762142, −5.391000739053855, −4.589075712809548, −3.957566719707330, −3.471921694750173, −2.672572808915003, −1.794142432832619, −0.8626349737607656, 0,
0.8626349737607656, 1.794142432832619, 2.672572808915003, 3.471921694750173, 3.957566719707330, 4.589075712809548, 5.391000739053855, 6.162577587762142, 6.530357370982660, 6.777888900560858, 7.552272640552805, 8.475853762791593, 8.966798364760700, 9.384660244495443, 10.08716942994901, 10.48979757200102, 11.04586125940362, 11.76022117635415, 12.22218396546880, 12.61370143947619, 13.17251665501460, 13.74156662388855, 14.18365637101948, 14.95555110349994, 15.45746563052728
