| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 2·11-s − 15-s − 4·17-s + 19-s + 21-s − 3·23-s − 4·25-s + 27-s − 3·29-s − 5·31-s + 2·33-s − 35-s − 8·37-s + 10·41-s − 43-s − 45-s + 13·47-s + 49-s − 4·51-s + 53-s − 2·55-s + 57-s − 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.603·11-s − 0.258·15-s − 0.970·17-s + 0.229·19-s + 0.218·21-s − 0.625·23-s − 4/5·25-s + 0.192·27-s − 0.557·29-s − 0.898·31-s + 0.348·33-s − 0.169·35-s − 1.31·37-s + 1.56·41-s − 0.152·43-s − 0.149·45-s + 1.89·47-s + 1/7·49-s − 0.560·51-s + 0.137·53-s − 0.269·55-s + 0.132·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 227136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.23025875417750, −12.61922532035542, −12.32189304479875, −11.74826863816279, −11.35190780851340, −10.85161937680069, −10.49564163892857, −9.760729521293315, −9.361905810851974, −8.947918442528237, −8.498402916894726, −8.015365941512231, −7.412017223537260, −7.224005622569686, −6.564440946134472, −5.966937467923159, −5.482059162578418, −4.860835532051947, −4.193359999229064, −3.865219040859610, −3.516954366718898, −2.579637792603168, −2.156734879451396, −1.617323125797071, −0.8170357584350549, 0,
0.8170357584350549, 1.617323125797071, 2.156734879451396, 2.579637792603168, 3.516954366718898, 3.865219040859610, 4.193359999229064, 4.860835532051947, 5.482059162578418, 5.966937467923159, 6.564440946134472, 7.224005622569686, 7.412017223537260, 8.015365941512231, 8.498402916894726, 8.947918442528237, 9.361905810851974, 9.760729521293315, 10.49564163892857, 10.85161937680069, 11.35190780851340, 11.74826863816279, 12.32189304479875, 12.61922532035542, 13.23025875417750
