| L(s) = 1 | + 0.285·2-s + 8.01·3-s − 7.91·4-s + 1.18·5-s + 2.29·6-s + 7·7-s − 4.55·8-s + 37.2·9-s + 0.337·10-s + 31.7·11-s − 63.4·12-s + 72.6·13-s + 2.00·14-s + 9.46·15-s + 62.0·16-s + 20.0·17-s + 10.6·18-s + 13.6·19-s − 9.35·20-s + 56.0·21-s + 9.06·22-s + 23·23-s − 36.4·24-s − 123.·25-s + 20.7·26-s + 81.7·27-s − 55.4·28-s + ⋯ |
| L(s) = 1 | + 0.101·2-s + 1.54·3-s − 0.989·4-s + 0.105·5-s + 0.155·6-s + 0.377·7-s − 0.201·8-s + 1.37·9-s + 0.0106·10-s + 0.869·11-s − 1.52·12-s + 1.54·13-s + 0.0382·14-s + 0.163·15-s + 0.969·16-s + 0.285·17-s + 0.139·18-s + 0.164·19-s − 0.104·20-s + 0.582·21-s + 0.0878·22-s + 0.208·23-s − 0.310·24-s − 0.988·25-s + 0.156·26-s + 0.582·27-s − 0.374·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.672790324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.672790324\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 0.285T + 8T^{2} \) |
| 3 | \( 1 - 8.01T + 27T^{2} \) |
| 5 | \( 1 - 1.18T + 125T^{2} \) |
| 11 | \( 1 - 31.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 72.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.6T + 6.85e3T^{2} \) |
| 29 | \( 1 + 184.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 38.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 275.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 369.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 146.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 28.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + 159.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 271.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 851.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 526.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 518.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 945.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 291.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 186.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.59e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.49e3T + 9.12e5T^{2} \) |
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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
−12.85760214257950627200014065247, −11.45884063599040712149674431944, −10.01181285025805769262216190827, −9.040207429890701795560356132193, −8.570084542331183782442244086402, −7.54069714715377030393993458242, −5.83495073452518751299726959879, −4.16331727248465649255993031164, −3.40486653860510279770925765555, −1.50979772384424093941421548204,
1.50979772384424093941421548204, 3.40486653860510279770925765555, 4.16331727248465649255993031164, 5.83495073452518751299726959879, 7.54069714715377030393993458242, 8.570084542331183782442244086402, 9.040207429890701795560356132193, 10.01181285025805769262216190827, 11.45884063599040712149674431944, 12.85760214257950627200014065247
