| L(s) = 1 | − 18.3·5-s + 14.9·7-s − 50.8·11-s − 26.1·13-s − 93.7·17-s + 71.2·19-s + 146.·23-s + 210.·25-s − 38.4·29-s + 11.2·31-s − 274.·35-s − 426.·37-s + 124.·41-s + 309.·43-s − 16.3·47-s − 118.·49-s + 529.·53-s + 931.·55-s + 439.·59-s − 54.0·61-s + 478.·65-s + 445.·67-s − 3.34·71-s + 820.·73-s − 761.·77-s + 133.·79-s − 265.·83-s + ⋯ |
| L(s) = 1 | − 1.63·5-s + 0.808·7-s − 1.39·11-s − 0.557·13-s − 1.33·17-s + 0.860·19-s + 1.32·23-s + 1.68·25-s − 0.246·29-s + 0.0651·31-s − 1.32·35-s − 1.89·37-s + 0.472·41-s + 1.09·43-s − 0.0506·47-s − 0.345·49-s + 1.37·53-s + 2.28·55-s + 0.969·59-s − 0.113·61-s + 0.913·65-s + 0.812·67-s − 0.00558·71-s + 1.31·73-s − 1.12·77-s + 0.190·79-s − 0.350·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.023192962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023192962\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 18.3T + 125T^{2} \) |
| 7 | \( 1 - 14.9T + 343T^{2} \) |
| 11 | \( 1 + 50.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 93.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 71.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 146.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 38.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 11.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 426.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 124.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 309.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 16.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 529.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 439.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 54.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 445.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.34T + 3.57e5T^{2} \) |
| 73 | \( 1 - 820.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 133.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 265.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.21e3T + 9.12e5T^{2} \) |
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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
−10.45301907682504733424917077717, −9.052627830862499202740030447402, −8.285001445019007222963580644108, −7.53698382099017558581109198809, −6.99728982754416217101129969922, −5.23894681416639679854797842884, −4.69990692229824436503471586288, −3.56268006568381485611844177374, −2.39727911852881652477850791702, −0.57338513987383176429241157455,
0.57338513987383176429241157455, 2.39727911852881652477850791702, 3.56268006568381485611844177374, 4.69990692229824436503471586288, 5.23894681416639679854797842884, 6.99728982754416217101129969922, 7.53698382099017558581109198809, 8.285001445019007222963580644108, 9.052627830862499202740030447402, 10.45301907682504733424917077717
