| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s − 11-s − 2·13-s + 15-s + 17-s − 19-s − 3·21-s − 6·23-s + 25-s + 27-s + 7·29-s + 10·31-s − 33-s − 3·35-s − 3·37-s − 2·39-s − 9·41-s − 8·43-s + 45-s − 3·47-s + 2·49-s + 51-s + 5·53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.258·15-s + 0.242·17-s − 0.229·19-s − 0.654·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.29·29-s + 1.79·31-s − 0.174·33-s − 0.507·35-s − 0.493·37-s − 0.320·39-s − 1.40·41-s − 1.21·43-s + 0.149·45-s − 0.437·47-s + 2/7·49-s + 0.140·51-s + 0.686·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.253362864612898826269052046918, −7.28561394922153805502201882620, −6.57064677286742119426049680215, −6.03503285076530914393593903837, −5.00954547383326488956220433140, −4.21628299339007278641591686994, −3.14861508379813796130333238374, −2.68300012496084197299053550963, −1.55859119120976698147361255770, 0,
1.55859119120976698147361255770, 2.68300012496084197299053550963, 3.14861508379813796130333238374, 4.21628299339007278641591686994, 5.00954547383326488956220433140, 6.03503285076530914393593903837, 6.57064677286742119426049680215, 7.28561394922153805502201882620, 8.253362864612898826269052046918
