| L(s) = 1 | + 5.09·2-s + 17.9·4-s + 18.5·5-s + 50.5·8-s + 94.7·10-s − 1.05·11-s + 58.7·13-s + 113.·16-s − 43.7·17-s − 131.·19-s + 333.·20-s − 5.37·22-s + 161.·23-s + 220.·25-s + 299.·26-s + 64.0·29-s − 55.9·31-s + 175.·32-s − 222.·34-s + 296.·37-s − 671.·38-s + 940.·40-s − 80.6·41-s − 134.·43-s − 18.9·44-s + 821.·46-s − 233.·47-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.24·4-s + 1.66·5-s + 2.23·8-s + 2.99·10-s − 0.0289·11-s + 1.25·13-s + 1.78·16-s − 0.624·17-s − 1.59·19-s + 3.72·20-s − 0.0521·22-s + 1.46·23-s + 1.76·25-s + 2.25·26-s + 0.409·29-s − 0.324·31-s + 0.971·32-s − 1.12·34-s + 1.31·37-s − 2.86·38-s + 3.71·40-s − 0.307·41-s − 0.476·43-s − 0.0648·44-s + 2.63·46-s − 0.724·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.996659253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.996659253\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 5.09T + 8T^{2} \) |
| 5 | \( 1 - 18.5T + 125T^{2} \) |
| 11 | \( 1 + 1.05T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 43.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 131.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 64.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 55.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 296.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 80.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 134.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 233.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 387.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 722.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 388.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 730.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 685.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 275.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 854.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 922.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 301.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 913.T + 9.12e5T^{2} \) |
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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
−9.268247589961891059992762740321, −8.523993764278241227193538157563, −7.04483946562513554441110443400, −6.28265691320685192678566816145, −5.99851798745741985321789458927, −5.02305550647267065062166447953, −4.29855540277592271752046942248, −3.13638320119988916412731000168, −2.29482140352078608693919621895, −1.42524946360190856971813238028,
1.42524946360190856971813238028, 2.29482140352078608693919621895, 3.13638320119988916412731000168, 4.29855540277592271752046942248, 5.02305550647267065062166447953, 5.99851798745741985321789458927, 6.28265691320685192678566816145, 7.04483946562513554441110443400, 8.523993764278241227193538157563, 9.268247589961891059992762740321
