| L(s) = 1 | + 7.49·2-s + 9·3-s + 24.1·4-s + 30.9·5-s + 67.4·6-s − 127.·7-s − 58.5·8-s + 81·9-s + 232.·10-s + 403.·11-s + 217.·12-s + 25.4·13-s − 953.·14-s + 278.·15-s − 1.21e3·16-s + 607.·18-s − 621.·19-s + 749.·20-s − 1.14e3·21-s + 3.02e3·22-s − 4.54e3·23-s − 526.·24-s − 2.16e3·25-s + 191.·26-s + 729·27-s − 3.07e3·28-s + 1.66e3·29-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.577·3-s + 0.756·4-s + 0.553·5-s + 0.765·6-s − 0.980·7-s − 0.323·8-s + 0.333·9-s + 0.733·10-s + 1.00·11-s + 0.436·12-s + 0.0418·13-s − 1.29·14-s + 0.319·15-s − 1.18·16-s + 0.441·18-s − 0.394·19-s + 0.418·20-s − 0.566·21-s + 1.33·22-s − 1.79·23-s − 0.186·24-s − 0.693·25-s + 0.0554·26-s + 0.192·27-s − 0.741·28-s + 0.368·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 867 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 7.49T + 32T^{2} \) |
| 5 | \( 1 - 30.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 127.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 403.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 25.4T + 3.71e5T^{2} \) |
| 19 | \( 1 + 621.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.54e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 156.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.23e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 3.63e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.35e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.17e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.81e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.23e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.93e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.43e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.14e5T + 8.58e9T^{2} \) |
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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
−9.193344974266842264701477524708, −8.158924078467990611035305494534, −6.83094511256754386267826383158, −6.26848098282254685968012937965, −5.52343863365490055337462918185, −4.21913210524043266318315570997, −3.71506609413760927245447075049, −2.72513713557794195029726002139, −1.73987433520344027445501669790, 0,
1.73987433520344027445501669790, 2.72513713557794195029726002139, 3.71506609413760927245447075049, 4.21913210524043266318315570997, 5.52343863365490055337462918185, 6.26848098282254685968012937965, 6.83094511256754386267826383158, 8.158924078467990611035305494534, 9.193344974266842264701477524708
