| L(s) = 1 | − 22·3-s − 25·5-s − 218·7-s + 241·9-s + 480·11-s − 622·13-s + 550·15-s + 186·17-s + 1.20e3·19-s + 4.79e3·21-s + 3.18e3·23-s + 625·25-s + 44·27-s + 5.52e3·29-s − 9.35e3·31-s − 1.05e4·33-s + 5.45e3·35-s + 5.61e3·37-s + 1.36e4·39-s − 1.43e4·41-s + 370·43-s − 6.02e3·45-s − 1.61e4·47-s + 3.07e4·49-s − 4.09e3·51-s − 4.37e3·53-s − 1.20e4·55-s + ⋯ |
| L(s) = 1 | − 1.41·3-s − 0.447·5-s − 1.68·7-s + 0.991·9-s + 1.19·11-s − 1.02·13-s + 0.631·15-s + 0.156·17-s + 0.765·19-s + 2.37·21-s + 1.25·23-s + 1/5·25-s + 0.0116·27-s + 1.22·29-s − 1.74·31-s − 1.68·33-s + 0.752·35-s + 0.674·37-s + 1.44·39-s − 1.33·41-s + 0.0305·43-s − 0.443·45-s − 1.06·47-s + 1.82·49-s − 0.220·51-s − 0.213·53-s − 0.534·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6230367624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6230367624\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 + 22 T + p^{5} T^{2} \) |
| 7 | \( 1 + 218 T + p^{5} T^{2} \) |
| 11 | \( 1 - 480 T + p^{5} T^{2} \) |
| 13 | \( 1 + 622 T + p^{5} T^{2} \) |
| 17 | \( 1 - 186 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1204 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3186 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5526 T + p^{5} T^{2} \) |
| 31 | \( 1 + 9356 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5618 T + p^{5} T^{2} \) |
| 41 | \( 1 + 14394 T + p^{5} T^{2} \) |
| 43 | \( 1 - 370 T + p^{5} T^{2} \) |
| 47 | \( 1 + 16146 T + p^{5} T^{2} \) |
| 53 | \( 1 + 4374 T + p^{5} T^{2} \) |
| 59 | \( 1 - 11748 T + p^{5} T^{2} \) |
| 61 | \( 1 - 13202 T + p^{5} T^{2} \) |
| 67 | \( 1 - 11542 T + p^{5} T^{2} \) |
| 71 | \( 1 - 29532 T + p^{5} T^{2} \) |
| 73 | \( 1 - 33698 T + p^{5} T^{2} \) |
| 79 | \( 1 + 31208 T + p^{5} T^{2} \) |
| 83 | \( 1 - 38466 T + p^{5} T^{2} \) |
| 89 | \( 1 - 119514 T + p^{5} T^{2} \) |
| 97 | \( 1 - 94658 T + p^{5} T^{2} \) |
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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
−12.99180423072000906025100238564, −12.20503587493974279058603668291, −11.43546142321154164328509956893, −10.14750686889858341477658672807, −9.210234300283595616961731312476, −7.09431922732523096374722704248, −6.39412849979514070350775092831, −5.04152291009129108696479856611, −3.39440251294930143411675588113, −0.62677463919458102093413935872,
0.62677463919458102093413935872, 3.39440251294930143411675588113, 5.04152291009129108696479856611, 6.39412849979514070350775092831, 7.09431922732523096374722704248, 9.210234300283595616961731312476, 10.14750686889858341477658672807, 11.43546142321154164328509956893, 12.20503587493974279058603668291, 12.99180423072000906025100238564
