| L(s) = 1 | − 3.27e4·4-s + 1.24e6·7-s + 3.97e8·13-s + 1.07e9·16-s + 7.70e9·19-s − 3.05e10·25-s − 4.07e10·28-s − 2.13e11·31-s + 1.09e12·37-s + 1.44e12·43-s − 3.19e12·49-s − 1.30e13·52-s + 4.02e13·61-s − 3.51e13·64-s + 9.90e13·67-s + 9.01e12·73-s − 2.52e14·76-s − 8.86e13·79-s + 4.95e14·91-s + 1.03e15·97-s + 1.00e15·100-s − 2.10e15·103-s − 3.78e15·109-s + 1.33e15·112-s + ⋯ |
| L(s) = 1 | − 4-s + 0.571·7-s + 1.75·13-s + 16-s + 1.97·19-s − 25-s − 0.571·28-s − 1.39·31-s + 1.88·37-s + 0.808·43-s − 0.673·49-s − 1.75·52-s + 1.63·61-s − 64-s + 1.99·67-s + 0.0955·73-s − 1.97·76-s − 0.519·79-s + 1.00·91-s + 1.30·97-s + 100-s − 1.68·103-s − 1.98·109-s + 0.571·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(1.604911516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.604911516\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + p^{15} T^{2} \) |
| 5 | \( 1 + p^{15} T^{2} \) |
| 7 | \( 1 - 1244900 T + p^{15} T^{2} \) |
| 11 | \( 1 + p^{15} T^{2} \) |
| 13 | \( 1 - 397771850 T + p^{15} T^{2} \) |
| 17 | \( 1 + p^{15} T^{2} \) |
| 19 | \( 1 - 7700827736 T + p^{15} T^{2} \) |
| 23 | \( 1 + p^{15} T^{2} \) |
| 29 | \( 1 + p^{15} T^{2} \) |
| 31 | \( 1 + 213681227452 T + p^{15} T^{2} \) |
| 37 | \( 1 - 1090158909950 T + p^{15} T^{2} \) |
| 41 | \( 1 + p^{15} T^{2} \) |
| 43 | \( 1 - 1440654152600 T + p^{15} T^{2} \) |
| 47 | \( 1 + p^{15} T^{2} \) |
| 53 | \( 1 + p^{15} T^{2} \) |
| 59 | \( 1 + p^{15} T^{2} \) |
| 61 | \( 1 - 40241378988902 T + p^{15} T^{2} \) |
| 67 | \( 1 - 99059017336400 T + p^{15} T^{2} \) |
| 71 | \( 1 + p^{15} T^{2} \) |
| 73 | \( 1 - 9014812804550 T + p^{15} T^{2} \) |
| 79 | \( 1 + 88692309079036 T + p^{15} T^{2} \) |
| 83 | \( 1 + p^{15} T^{2} \) |
| 89 | \( 1 + p^{15} T^{2} \) |
| 97 | \( 1 - 1035097921427150 T + p^{15} T^{2} \) |
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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
−17.79780110147631823373253961061, −16.05509556515661687431560865293, −14.29372081226246405241536938693, −13.20865134886399636196772684840, −11.32099239754414778292205224216, −9.450981873958175511897362273980, −7.998401995954092643666642278857, −5.54774701736009056273507675234, −3.78383245926424319049095593166, −1.06787213504324241938934706428,
1.06787213504324241938934706428, 3.78383245926424319049095593166, 5.54774701736009056273507675234, 7.998401995954092643666642278857, 9.450981873958175511897362273980, 11.32099239754414778292205224216, 13.20865134886399636196772684840, 14.29372081226246405241536938693, 16.05509556515661687431560865293, 17.79780110147631823373253961061
