| L(s) = 1 | + 2·3-s − 2·5-s + 7-s + 9-s − 11-s + 13-s − 4·15-s + 3·17-s + 4·19-s + 2·21-s − 23-s − 25-s − 4·27-s − 3·29-s + 3·31-s − 2·33-s − 2·35-s + 3·37-s + 2·39-s − 3·41-s + 4·43-s − 2·45-s + 49-s + 6·51-s + 10·53-s + 2·55-s + 8·57-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 1.03·15-s + 0.727·17-s + 0.917·19-s + 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.769·27-s − 0.557·29-s + 0.538·31-s − 0.348·33-s − 0.338·35-s + 0.493·37-s + 0.320·39-s − 0.468·41-s + 0.609·43-s − 0.298·45-s + 1/7·49-s + 0.840·51-s + 1.37·53-s + 0.269·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.808618304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.808618304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 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
−15.68955435223174, −15.38706304263338, −14.74627558803832, −14.28315788939059, −13.83567260447057, −13.23953702039838, −12.66903711017160, −11.82406046332686, −11.60500636068323, −10.94200166358457, −10.09646269564818, −9.659669399754188, −8.961292399117934, −8.271982072991423, −8.058396996093625, −7.428701543922303, −6.948666967652272, −5.779060940927080, −5.391782048493711, −4.343759980854136, −3.853551588868503, −3.206365553785369, −2.594854896018125, −1.705667515690041, −0.6753504112401442,
0.6753504112401442, 1.705667515690041, 2.594854896018125, 3.206365553785369, 3.853551588868503, 4.343759980854136, 5.391782048493711, 5.779060940927080, 6.948666967652272, 7.428701543922303, 8.058396996093625, 8.271982072991423, 8.961292399117934, 9.659669399754188, 10.09646269564818, 10.94200166358457, 11.60500636068323, 11.82406046332686, 12.66903711017160, 13.23953702039838, 13.83567260447057, 14.28315788939059, 14.74627558803832, 15.38706304263338, 15.68955435223174
