| L(s) = 1 | − 0.585·3-s + 5-s − 4.82·7-s − 2.65·9-s − 2·11-s + 2.24·13-s − 0.585·15-s − 4.82·17-s − 19-s + 2.82·21-s − 6·23-s + 25-s + 3.31·27-s + 10.4·29-s − 1.17·31-s + 1.17·33-s − 4.82·35-s − 10.2·37-s − 1.31·39-s − 7.65·41-s − 0.828·43-s − 2.65·45-s + 0.828·47-s + 16.3·49-s + 2.82·51-s − 5.07·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.338·3-s + 0.447·5-s − 1.82·7-s − 0.885·9-s − 0.603·11-s + 0.621·13-s − 0.151·15-s − 1.17·17-s − 0.229·19-s + 0.617·21-s − 1.25·23-s + 0.200·25-s + 0.637·27-s + 1.94·29-s − 0.210·31-s + 0.203·33-s − 0.816·35-s − 1.68·37-s − 0.210·39-s − 1.19·41-s − 0.126·43-s − 0.396·45-s + 0.120·47-s + 2.33·49-s + 0.396·51-s − 0.696·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.585T + 3T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 + 4.82T + 17T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 7.65T + 41T^{2} \) |
| 43 | \( 1 + 0.828T + 43T^{2} \) |
| 47 | \( 1 - 0.828T + 47T^{2} \) |
| 53 | \( 1 + 5.07T + 53T^{2} \) |
| 59 | \( 1 - 2.34T + 59T^{2} \) |
| 61 | \( 1 + 2.34T + 61T^{2} \) |
| 67 | \( 1 + 0.585T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 9.65T + 79T^{2} \) |
| 83 | \( 1 - 9.31T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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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
−10.68548639853506772334618426967, −10.13051823443032030003347563002, −9.087019736397755837878264379104, −8.306331735166356284250669934194, −6.61046132542093407854307811113, −6.32492184653728948339791911819, −5.17790341794470007537249143915, −3.60239909983863243422565985455, −2.51380771504828713069998884601, 0,
2.51380771504828713069998884601, 3.60239909983863243422565985455, 5.17790341794470007537249143915, 6.32492184653728948339791911819, 6.61046132542093407854307811113, 8.306331735166356284250669934194, 9.087019736397755837878264379104, 10.13051823443032030003347563002, 10.68548639853506772334618426967
