| L(s) = 1 | + 128·2-s + 918·3-s + 1.63e4·4-s + 1.17e5·6-s + 9.53e5·7-s + 2.09e6·8-s − 1.35e7·9-s + 1.77e7·11-s + 1.50e7·12-s − 1.40e8·13-s + 1.22e8·14-s + 2.68e8·16-s − 2.99e9·17-s − 1.72e9·18-s + 3.25e9·19-s + 8.75e8·21-s + 2.27e9·22-s − 6.77e9·23-s + 1.92e9·24-s − 1.79e10·26-s − 2.55e10·27-s + 1.56e10·28-s − 7.34e9·29-s − 1.15e11·31-s + 3.43e10·32-s + 1.63e10·33-s − 3.83e11·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.242·3-s + 1/2·4-s + 0.171·6-s + 0.437·7-s + 0.353·8-s − 0.941·9-s + 0.275·11-s + 0.121·12-s − 0.621·13-s + 0.309·14-s + 1/4·16-s − 1.77·17-s − 0.665·18-s + 0.835·19-s + 0.106·21-s + 0.194·22-s − 0.414·23-s + 0.0856·24-s − 0.439·26-s − 0.470·27-s + 0.218·28-s − 0.0790·29-s − 0.753·31-s + 0.176·32-s + 0.0666·33-s − 1.25·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{7} T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 34 p^{3} T + p^{15} T^{2} \) |
| 7 | \( 1 - 136222 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 17783232 T + p^{15} T^{2} \) |
| 13 | \( 1 + 140533322 T + p^{15} T^{2} \) |
| 17 | \( 1 + 2998870746 T + p^{15} T^{2} \) |
| 19 | \( 1 - 3255852500 T + p^{15} T^{2} \) |
| 23 | \( 1 + 6774812202 T + p^{15} T^{2} \) |
| 29 | \( 1 + 7340322690 T + p^{15} T^{2} \) |
| 31 | \( 1 + 115428411388 T + p^{15} T^{2} \) |
| 37 | \( 1 + 150300986906 T + p^{15} T^{2} \) |
| 41 | \( 1 - 1841603525142 T + p^{15} T^{2} \) |
| 43 | \( 1 + 1510018315682 T + p^{15} T^{2} \) |
| 47 | \( 1 + 6093750843366 T + p^{15} T^{2} \) |
| 53 | \( 1 - 8267412829038 T + p^{15} T^{2} \) |
| 59 | \( 1 + 23516883061980 T + p^{15} T^{2} \) |
| 61 | \( 1 + 3135369104278 T + p^{15} T^{2} \) |
| 67 | \( 1 - 36030983954794 T + p^{15} T^{2} \) |
| 71 | \( 1 - 52169735384172 T + p^{15} T^{2} \) |
| 73 | \( 1 + 69977143684082 T + p^{15} T^{2} \) |
| 79 | \( 1 + 135317670906760 T + p^{15} T^{2} \) |
| 83 | \( 1 + 427456158822882 T + p^{15} T^{2} \) |
| 89 | \( 1 + 446581617299190 T + p^{15} T^{2} \) |
| 97 | \( 1 + 181247411845826 T + p^{15} T^{2} \) |
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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
−11.76731835840204252504205881247, −11.01732216711611564107430466197, −9.393173661368005280468247785748, −8.156635721703230584236300642315, −6.83030234412745767761725985376, −5.51693193538138289269016665429, −4.35663630803731575270248229811, −2.96199081929848571878847159308, −1.83724677289317233914658675907, 0,
1.83724677289317233914658675907, 2.96199081929848571878847159308, 4.35663630803731575270248229811, 5.51693193538138289269016665429, 6.83030234412745767761725985376, 8.156635721703230584236300642315, 9.393173661368005280468247785748, 11.01732216711611564107430466197, 11.76731835840204252504205881247
