| L(s) = 1 | + 0.304·3-s − 35.5·5-s − 49·7-s − 242.·9-s + 565.·11-s − 983.·13-s − 10.8·15-s + 200.·17-s + 828.·19-s − 14.9·21-s − 4.43e3·23-s − 1.86e3·25-s − 147.·27-s + 3.71e3·29-s − 992.·31-s + 172.·33-s + 1.74e3·35-s + 8.35e3·37-s − 299.·39-s − 1.34e4·41-s + 298.·43-s + 8.62e3·45-s + 1.87e4·47-s + 2.40e3·49-s + 60.8·51-s − 1.60e4·53-s − 2.00e4·55-s + ⋯ |
| L(s) = 1 | + 0.0195·3-s − 0.635·5-s − 0.377·7-s − 0.999·9-s + 1.40·11-s − 1.61·13-s − 0.0123·15-s + 0.167·17-s + 0.526·19-s − 0.00737·21-s − 1.74·23-s − 0.596·25-s − 0.0390·27-s + 0.820·29-s − 0.185·31-s + 0.0275·33-s + 0.240·35-s + 1.00·37-s − 0.0314·39-s − 1.25·41-s + 0.0246·43-s + 0.635·45-s + 1.23·47-s + 0.142·49-s + 0.00327·51-s − 0.784·53-s − 0.895·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.111193557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.111193557\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 49T \) |
| good | 3 | \( 1 - 0.304T + 243T^{2} \) |
| 5 | \( 1 + 35.5T + 3.12e3T^{2} \) |
| 11 | \( 1 - 565.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 983.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 200.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 828.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.43e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 992.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.35e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 298.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.87e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.60e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.19e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.29e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.11e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.47e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.26e4T + 8.58e9T^{2} \) |
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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
−10.09722694272163780500435925270, −9.484110496366488042071638998902, −8.434913414151653896298519983112, −7.58675218674520602490354732949, −6.59478390328425456077222121689, −5.61560310304633749081479944148, −4.36394503355686493807207108902, −3.41404759561890926887120510069, −2.20470176531966949214438510023, −0.52178033605158052394811031589,
0.52178033605158052394811031589, 2.20470176531966949214438510023, 3.41404759561890926887120510069, 4.36394503355686493807207108902, 5.61560310304633749081479944148, 6.59478390328425456077222121689, 7.58675218674520602490354732949, 8.434913414151653896298519983112, 9.484110496366488042071638998902, 10.09722694272163780500435925270
