L(s) = 1 | + 30.3·3-s + 100.·5-s − 49·7-s + 676.·9-s + 140.·11-s − 725.·13-s + 3.04e3·15-s − 365.·17-s + 202.·19-s − 1.48e3·21-s + 2.73e3·23-s + 6.95e3·25-s + 1.31e4·27-s − 2.17e3·29-s − 8.95e3·31-s + 4.25e3·33-s − 4.91e3·35-s + 2.60e3·37-s − 2.19e4·39-s + 1.42e4·41-s + 1.12e4·43-s + 6.79e4·45-s + 1.77e4·47-s + 2.40e3·49-s − 1.10e4·51-s − 2.59e4·53-s + 1.40e4·55-s + ⋯ |
L(s) = 1 | + 1.94·3-s + 1.79·5-s − 0.377·7-s + 2.78·9-s + 0.349·11-s − 1.19·13-s + 3.49·15-s − 0.306·17-s + 0.128·19-s − 0.735·21-s + 1.07·23-s + 2.22·25-s + 3.47·27-s − 0.480·29-s − 1.67·31-s + 0.680·33-s − 0.678·35-s + 0.312·37-s − 2.31·39-s + 1.32·41-s + 0.926·43-s + 5.00·45-s + 1.17·47-s + 0.142·49-s − 0.597·51-s − 1.26·53-s + 0.628·55-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
\(L(3)\) |
\(\approx\) |
\(6.396469266\) |
\(L(\frac12)\) |
\(\approx\) |
\(6.396469266\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 30.3T + 243T^{2} \) |
| 5 | \( 1 - 100.T + 3.12e3T^{2} \) |
| 11 | \( 1 - 140.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 725.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 365.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 202.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.73e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.17e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.95e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.42e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.12e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.59e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.76e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.93e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 854.T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.91e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.88e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.55e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.76e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.14e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.96e4T + 8.58e9T^{2} \) |
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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
−9.826625639943145665137765841128, −9.297195200760575119457915270731, −8.926463495479328924447188553538, −7.51286618139772598759450194646, −6.85930728161197779703455718506, −5.55329624771796622913512841619, −4.28260748774100532653656515142, −2.92836274895404229594944139727, −2.32071434533119790629198710139, −1.38107287485903568329374768848,
1.38107287485903568329374768848, 2.32071434533119790629198710139, 2.92836274895404229594944139727, 4.28260748774100532653656515142, 5.55329624771796622913512841619, 6.85930728161197779703455718506, 7.51286618139772598759450194646, 8.926463495479328924447188553538, 9.297195200760575119457915270731, 9.826625639943145665137765841128