| L(s) = 1 | + 27·3-s + 270·5-s + 1.11e3·7-s + 729·9-s − 5.72e3·11-s − 4.57e3·13-s + 7.29e3·15-s − 3.65e4·17-s + 5.17e4·19-s + 3.00e4·21-s + 2.22e4·23-s − 5.22e3·25-s + 1.96e4·27-s − 1.57e5·29-s − 1.03e5·31-s − 1.54e5·33-s + 3.00e5·35-s − 9.48e4·37-s − 1.23e5·39-s + 6.59e5·41-s − 7.57e4·43-s + 1.96e5·45-s + 4.05e5·47-s + 4.13e5·49-s − 9.87e5·51-s − 1.34e6·53-s − 1.54e6·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.965·5-s + 1.22·7-s + 1/3·9-s − 1.29·11-s − 0.576·13-s + 0.557·15-s − 1.80·17-s + 1.73·19-s + 0.707·21-s + 0.381·23-s − 0.0668·25-s + 0.192·27-s − 1.19·29-s − 0.626·31-s − 0.748·33-s + 1.18·35-s − 0.307·37-s − 0.333·39-s + 1.49·41-s − 0.145·43-s + 0.321·45-s + 0.569·47-s + 0.501·49-s − 1.04·51-s − 1.24·53-s − 1.25·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.935795292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.935795292\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| good | 5 | \( 1 - 54 p T + p^{7} T^{2} \) |
| 7 | \( 1 - 1112 T + p^{7} T^{2} \) |
| 11 | \( 1 + 5724 T + p^{7} T^{2} \) |
| 13 | \( 1 + 4570 T + p^{7} T^{2} \) |
| 17 | \( 1 + 36558 T + p^{7} T^{2} \) |
| 19 | \( 1 - 51740 T + p^{7} T^{2} \) |
| 23 | \( 1 - 22248 T + p^{7} T^{2} \) |
| 29 | \( 1 + 157194 T + p^{7} T^{2} \) |
| 31 | \( 1 + 103936 T + p^{7} T^{2} \) |
| 37 | \( 1 + 94834 T + p^{7} T^{2} \) |
| 41 | \( 1 - 659610 T + p^{7} T^{2} \) |
| 43 | \( 1 + 75772 T + p^{7} T^{2} \) |
| 47 | \( 1 - 405648 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1346274 T + p^{7} T^{2} \) |
| 59 | \( 1 + 1303884 T + p^{7} T^{2} \) |
| 61 | \( 1 - 30062 p T + p^{7} T^{2} \) |
| 67 | \( 1 - 1369388 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2714040 T + p^{7} T^{2} \) |
| 73 | \( 1 - 2868794 T + p^{7} T^{2} \) |
| 79 | \( 1 + 1129648 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5912028 T + p^{7} T^{2} \) |
| 89 | \( 1 + 897750 T + p^{7} T^{2} \) |
| 97 | \( 1 - 13719074 T + p^{7} T^{2} \) |
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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
−18.32086666196737037491929495256, −17.54800185750210380385075659803, −15.61160800833991202667733492757, −14.22751430239593901659851824046, −13.15163103672418303365702068758, −11.02623669318394053275884873432, −9.365380720505657791462489118007, −7.65138268663080674998293473808, −5.14970740096056688520754894161, −2.17027281891390998598518049871,
2.17027281891390998598518049871, 5.14970740096056688520754894161, 7.65138268663080674998293473808, 9.365380720505657791462489118007, 11.02623669318394053275884873432, 13.15163103672418303365702068758, 14.22751430239593901659851824046, 15.61160800833991202667733492757, 17.54800185750210380385075659803, 18.32086666196737037491929495256
