| L(s) = 1 | + 4.50·2-s + 5.08·3-s + 12.2·4-s − 5·5-s + 22.9·6-s − 0.616·7-s + 19.3·8-s − 1.09·9-s − 22.5·10-s − 8.63·11-s + 62.5·12-s + 6.44·13-s − 2.77·14-s − 25.4·15-s − 11.2·16-s − 17·17-s − 4.91·18-s + 7.96·19-s − 61.4·20-s − 3.13·21-s − 38.8·22-s + 66.1·23-s + 98.4·24-s + 25·25-s + 29.0·26-s − 142.·27-s − 7.58·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.979·3-s + 1.53·4-s − 0.447·5-s + 1.56·6-s − 0.0332·7-s + 0.854·8-s − 0.0404·9-s − 0.712·10-s − 0.236·11-s + 1.50·12-s + 0.137·13-s − 0.0530·14-s − 0.438·15-s − 0.175·16-s − 0.242·17-s − 0.0644·18-s + 0.0962·19-s − 0.687·20-s − 0.0326·21-s − 0.376·22-s + 0.599·23-s + 0.836·24-s + 0.200·25-s + 0.218·26-s − 1.01·27-s − 0.0511·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.946676989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.946676989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 - 4.50T + 8T^{2} \) |
| 3 | \( 1 - 5.08T + 27T^{2} \) |
| 7 | \( 1 + 0.616T + 343T^{2} \) |
| 11 | \( 1 + 8.63T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6.44T + 2.19e3T^{2} \) |
| 19 | \( 1 - 7.96T + 6.85e3T^{2} \) |
| 23 | \( 1 - 66.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 0.608T + 2.97e4T^{2} \) |
| 37 | \( 1 + 216.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 355.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 209.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 324.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 189.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 257.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 240.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 66.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 131.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.17e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 707.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.00e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.01e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 973.T + 9.12e5T^{2} \) |
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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
−13.84607403568569026553937743451, −12.96371228446560967973435590436, −11.96700065379150229602644614104, −10.87340721492894986045430752165, −9.131126785841765477617157557679, −7.909279614510233861968010838681, −6.51524584481201277119234762981, −5.02466628915192005655822693514, −3.70063376827350616544234187635, −2.62531486020739504467690275918,
2.62531486020739504467690275918, 3.70063376827350616544234187635, 5.02466628915192005655822693514, 6.51524584481201277119234762981, 7.909279614510233861968010838681, 9.131126785841765477617157557679, 10.87340721492894986045430752165, 11.96700065379150229602644614104, 12.96371228446560967973435590436, 13.84607403568569026553937743451
