| L(s) = 1 | + 6.55e4·2-s + 4.30e7·3-s + 4.29e9·4-s − 3.04e11·5-s + 2.82e12·6-s + 6.55e13·7-s + 2.81e14·8-s + 1.85e15·9-s − 1.99e16·10-s + 2.75e16·11-s + 1.84e17·12-s + 4.29e18·14-s − 1.30e19·15-s + 1.84e19·16-s + 1.21e20·18-s − 1.30e21·20-s + 2.82e21·21-s + 1.80e21·22-s + 1.21e22·24-s + 6.92e22·25-s + 7.97e22·27-s + 2.81e23·28-s − 3.02e23·29-s − 8.58e23·30-s − 6.19e23·31-s + 1.20e24·32-s + 1.18e24·33-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 1.99·5-s + 6-s + 1.97·7-s + 8-s + 9-s − 1.99·10-s + 0.599·11-s + 12-s + 1.97·14-s − 1.99·15-s + 16-s + 18-s − 1.99·20-s + 1.97·21-s + 0.599·22-s + 24-s + 2.97·25-s + 27-s + 1.97·28-s − 1.20·29-s − 1.99·30-s − 0.851·31-s + 32-s + 0.599·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{33}{2})\) |
\(\approx\) |
\(7.118725297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.118725297\) |
| \(L(17)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{16} T \) |
| 3 | \( 1 - p^{16} T \) |
| good | 5 | \( 1 + 304196466814 T + p^{32} T^{2} \) |
| 7 | \( 1 - 65581862265602 T + p^{32} T^{2} \) |
| 11 | \( 1 - 27525964704961922 T + p^{32} T^{2} \) |
| 13 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 17 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 19 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 23 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 29 | \( 1 + \)\(30\!\cdots\!58\)\( T + p^{32} T^{2} \) |
| 31 | \( 1 + \)\(61\!\cdots\!38\)\( T + p^{32} T^{2} \) |
| 37 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 41 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 43 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 47 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 53 | \( 1 - \)\(12\!\cdots\!42\)\( T + p^{32} T^{2} \) |
| 59 | \( 1 - \)\(91\!\cdots\!82\)\( T + p^{32} T^{2} \) |
| 61 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 67 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 71 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 73 | \( 1 - \)\(10\!\cdots\!22\)\( T + p^{32} T^{2} \) |
| 79 | \( 1 - \)\(44\!\cdots\!42\)\( T + p^{32} T^{2} \) |
| 83 | \( 1 + \)\(79\!\cdots\!38\)\( T + p^{32} T^{2} \) |
| 89 | \( ( 1 - p^{16} T )( 1 + p^{16} T ) \) |
| 97 | \( 1 + \)\(12\!\cdots\!58\)\( T + p^{32} T^{2} \) |
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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
−11.57429955188929230196793879199, −10.93521226820918686949332341657, −8.623911554896208475819303917478, −7.77806585843255776311173716024, −7.19134334611612509504751562372, −4.98517755288263147288217532216, −4.15642373394782806872863083793, −3.53084070175647313633774804540, −2.09185934629084513476160128340, −1.04074580829954154043944495003,
1.04074580829954154043944495003, 2.09185934629084513476160128340, 3.53084070175647313633774804540, 4.15642373394782806872863083793, 4.98517755288263147288217532216, 7.19134334611612509504751562372, 7.77806585843255776311173716024, 8.623911554896208475819303917478, 10.93521226820918686949332341657, 11.57429955188929230196793879199
