| L(s) = 1 | − 15·3-s + 19·5-s − 10·7-s − 18·9-s − 121·11-s + 1.14e3·13-s − 285·15-s + 686·17-s − 384·19-s + 150·21-s − 3.70e3·23-s − 2.76e3·25-s + 3.91e3·27-s + 5.42e3·29-s + 6.44e3·31-s + 1.81e3·33-s − 190·35-s − 1.20e4·37-s − 1.72e4·39-s − 1.52e3·41-s − 4.02e3·43-s − 342·45-s − 7.16e3·47-s − 1.67e4·49-s − 1.02e4·51-s + 2.98e4·53-s − 2.29e3·55-s + ⋯ |
| L(s) = 1 | − 0.962·3-s + 0.339·5-s − 0.0771·7-s − 0.0740·9-s − 0.301·11-s + 1.88·13-s − 0.327·15-s + 0.575·17-s − 0.244·19-s + 0.0742·21-s − 1.46·23-s − 0.884·25-s + 1.03·27-s + 1.19·29-s + 1.20·31-s + 0.290·33-s − 0.0262·35-s − 1.44·37-s − 1.81·39-s − 0.141·41-s − 0.332·43-s − 0.0251·45-s − 0.473·47-s − 0.994·49-s − 0.553·51-s + 1.46·53-s − 0.102·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.402158586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402158586\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 5 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 19 T + p^{5} T^{2} \) |
| 7 | \( 1 + 10 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1148 T + p^{5} T^{2} \) |
| 17 | \( 1 - 686 T + p^{5} T^{2} \) |
| 19 | \( 1 + 384 T + p^{5} T^{2} \) |
| 23 | \( 1 + 3709 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5424 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6443 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12063 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1528 T + p^{5} T^{2} \) |
| 43 | \( 1 + 4026 T + p^{5} T^{2} \) |
| 47 | \( 1 + 7168 T + p^{5} T^{2} \) |
| 53 | \( 1 - 29862 T + p^{5} T^{2} \) |
| 59 | \( 1 + 6461 T + p^{5} T^{2} \) |
| 61 | \( 1 - 16980 T + p^{5} T^{2} \) |
| 67 | \( 1 - 29999 T + p^{5} T^{2} \) |
| 71 | \( 1 + 31023 T + p^{5} T^{2} \) |
| 73 | \( 1 - 1924 T + p^{5} T^{2} \) |
| 79 | \( 1 + 65138 T + p^{5} T^{2} \) |
| 83 | \( 1 + 102714 T + p^{5} T^{2} \) |
| 89 | \( 1 - 17415 T + p^{5} T^{2} \) |
| 97 | \( 1 - 66905 T + p^{5} T^{2} \) |
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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
−10.05499657251013822977265693954, −8.628844629049000963468271476091, −8.148970473999582096743533389222, −6.69271416025865844105496504440, −6.04795445252192341060801028039, −5.45312806199095787377610092797, −4.25040718987858735335098083698, −3.15663569742535226950396922649, −1.70508900018107899854435541637, −0.59623933633395996698395458607,
0.59623933633395996698395458607, 1.70508900018107899854435541637, 3.15663569742535226950396922649, 4.25040718987858735335098083698, 5.45312806199095787377610092797, 6.04795445252192341060801028039, 6.69271416025865844105496504440, 8.148970473999582096743533389222, 8.628844629049000963468271476091, 10.05499657251013822977265693954
