| L(s) = 1 | + 3-s − 5-s − 7-s − 2·9-s − 4·13-s − 15-s + 17-s − 4·19-s − 21-s − 2·23-s − 4·25-s − 5·27-s + 6·29-s − 7·31-s + 35-s − 2·37-s − 4·39-s − 6·41-s − 7·43-s + 2·45-s + 8·47-s + 49-s + 51-s − 53-s − 4·57-s + 10·59-s + 61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.10·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s − 0.218·21-s − 0.417·23-s − 4/5·25-s − 0.962·27-s + 1.11·29-s − 1.25·31-s + 0.169·35-s − 0.328·37-s − 0.640·39-s − 0.937·41-s − 1.06·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.140·51-s − 0.137·53-s − 0.529·57-s + 1.30·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−13.09171007105566, −12.66299834141992, −12.28121029061123, −11.79510838446020, −11.36736105185054, −10.89643999676257, −10.21505257178090, −9.866818436845284, −9.510130752777368, −8.802247921036986, −8.425268444060288, −8.107113368947601, −7.562714731291862, −6.885319562173676, −6.743359753678393, −5.893417283820426, −5.431156668649225, −5.017046584723138, −4.178611852532547, −3.866737012258185, −3.293400587193738, −2.691714538577069, −2.225044563085222, −1.694758487553754, −0.5688120681415412, 0,
0.5688120681415412, 1.694758487553754, 2.225044563085222, 2.691714538577069, 3.293400587193738, 3.866737012258185, 4.178611852532547, 5.017046584723138, 5.431156668649225, 5.893417283820426, 6.743359753678393, 6.885319562173676, 7.562714731291862, 8.107113368947601, 8.425268444060288, 8.802247921036986, 9.510130752777368, 9.866818436845284, 10.21505257178090, 10.89643999676257, 11.36736105185054, 11.79510838446020, 12.28121029061123, 12.66299834141992, 13.09171007105566
