| L(s) = 1 | + 4.04e4·3-s + 1.95e6·5-s − 2.02e8·7-s + 4.72e8·9-s − 4.72e9·11-s − 4.99e10·13-s + 7.89e10·15-s − 6.49e10·17-s + 5.24e11·19-s − 8.18e12·21-s − 1.87e12·23-s + 3.81e12·25-s − 2.78e13·27-s + 7.67e13·29-s − 3.34e13·31-s − 1.91e14·33-s − 3.95e14·35-s − 2.32e14·37-s − 2.02e15·39-s − 8.39e14·41-s + 6.08e15·43-s + 9.22e14·45-s − 7.49e15·47-s + 2.95e16·49-s − 2.62e15·51-s + 4.53e16·53-s − 9.23e15·55-s + ⋯ |
| L(s) = 1 | + 1.18·3-s + 0.447·5-s − 1.89·7-s + 0.406·9-s − 0.604·11-s − 1.30·13-s + 0.530·15-s − 0.132·17-s + 0.373·19-s − 2.24·21-s − 0.217·23-s + 0.199·25-s − 0.703·27-s + 0.982·29-s − 0.227·31-s − 0.716·33-s − 0.848·35-s − 0.294·37-s − 1.54·39-s − 0.400·41-s + 1.84·43-s + 0.181·45-s − 0.977·47-s + 2.59·49-s − 0.157·51-s + 1.88·53-s − 0.270·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(1.966923122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966923122\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 1.95e6T \) |
| good | 3 | \( 1 - 4.04e4T + 1.16e9T^{2} \) |
| 7 | \( 1 + 2.02e8T + 1.13e16T^{2} \) |
| 11 | \( 1 + 4.72e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 4.99e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 6.49e10T + 2.39e23T^{2} \) |
| 19 | \( 1 - 5.24e11T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.87e12T + 7.46e25T^{2} \) |
| 29 | \( 1 - 7.67e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 3.34e13T + 2.16e28T^{2} \) |
| 37 | \( 1 + 2.32e14T + 6.24e29T^{2} \) |
| 41 | \( 1 + 8.39e14T + 4.39e30T^{2} \) |
| 43 | \( 1 - 6.08e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 7.49e15T + 5.88e31T^{2} \) |
| 53 | \( 1 - 4.53e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 5.12e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 1.23e17T + 8.34e33T^{2} \) |
| 67 | \( 1 + 3.13e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 2.04e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 6.98e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.01e18T + 1.13e36T^{2} \) |
| 83 | \( 1 - 3.26e18T + 2.90e36T^{2} \) |
| 89 | \( 1 - 1.84e18T + 1.09e37T^{2} \) |
| 97 | \( 1 - 8.70e18T + 5.60e37T^{2} \) |
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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
−10.28191988775793137751150587601, −9.662011209105750143267668552979, −8.870250025595563222524736513254, −7.58541652890597167073063709593, −6.61396884576684183880822298183, −5.38592206917495614923188796723, −3.80468963425162704512535546809, −2.79811185849136473707759139222, −2.34482517568439988175236715121, −0.52364170983998490282369920732,
0.52364170983998490282369920732, 2.34482517568439988175236715121, 2.79811185849136473707759139222, 3.80468963425162704512535546809, 5.38592206917495614923188796723, 6.61396884576684183880822298183, 7.58541652890597167073063709593, 8.870250025595563222524736513254, 9.662011209105750143267668552979, 10.28191988775793137751150587601
