| L(s) = 1 | − 3-s + 7-s + 9-s − 13-s − 5·19-s − 21-s − 6·23-s − 27-s + 3·29-s − 4·31-s + 7·37-s + 39-s + 12·41-s + 2·43-s − 3·47-s + 49-s − 6·53-s + 5·57-s + 3·59-s − 2·61-s + 63-s + 67-s + 6·69-s + 12·71-s − 7·73-s + 4·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.277·13-s − 1.14·19-s − 0.218·21-s − 1.25·23-s − 0.192·27-s + 0.557·29-s − 0.718·31-s + 1.15·37-s + 0.160·39-s + 1.87·41-s + 0.304·43-s − 0.437·47-s + 1/7·49-s − 0.824·53-s + 0.662·57-s + 0.390·59-s − 0.256·61-s + 0.125·63-s + 0.122·67-s + 0.722·69-s + 1.42·71-s − 0.819·73-s + 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.570823674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.570823674\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.74276995831028, −12.52922126587198, −11.75067565753884, −11.54851653674252, −10.98797736600656, −10.55274488229681, −10.22723951770764, −9.497876958637641, −9.300925026430388, −8.578272740160563, −8.058564080001225, −7.722138925620697, −7.217290199143327, −6.503386469625777, −6.201421147489068, −5.747413104051237, −5.165188951771243, −4.546834960527945, −4.242415669958420, −3.716203952357417, −2.913724234418529, −2.240034319722186, −1.887799883291402, −1.025373753014226, −0.3953389571783909,
0.3953389571783909, 1.025373753014226, 1.887799883291402, 2.240034319722186, 2.913724234418529, 3.716203952357417, 4.242415669958420, 4.546834960527945, 5.165188951771243, 5.747413104051237, 6.201421147489068, 6.503386469625777, 7.217290199143327, 7.722138925620697, 8.058564080001225, 8.578272740160563, 9.300925026430388, 9.497876958637641, 10.22723951770764, 10.55274488229681, 10.98797736600656, 11.54851653674252, 11.75067565753884, 12.52922126587198, 12.74276995831028
