| L(s) = 1 | + 3·3-s + 4·5-s + 9·9-s + 20·11-s − 4·13-s + 12·15-s + 24·17-s − 44·19-s − 72·23-s − 109·25-s + 27·27-s − 38·29-s − 184·31-s + 60·33-s − 30·37-s − 12·39-s − 216·41-s + 164·43-s + 36·45-s − 520·47-s + 72·51-s − 146·53-s + 80·55-s − 132·57-s − 460·59-s + 628·61-s − 16·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.357·5-s + 1/3·9-s + 0.548·11-s − 0.0853·13-s + 0.206·15-s + 0.342·17-s − 0.531·19-s − 0.652·23-s − 0.871·25-s + 0.192·27-s − 0.243·29-s − 1.06·31-s + 0.316·33-s − 0.133·37-s − 0.0492·39-s − 0.822·41-s + 0.581·43-s + 0.119·45-s − 1.61·47-s + 0.197·51-s − 0.378·53-s + 0.196·55-s − 0.306·57-s − 1.01·59-s + 1.31·61-s − 0.0305·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 T + p^{3} T^{2} \) |
| 17 | \( 1 - 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 + 38 T + p^{3} T^{2} \) |
| 31 | \( 1 + 184 T + p^{3} T^{2} \) |
| 37 | \( 1 + 30 T + p^{3} T^{2} \) |
| 41 | \( 1 + 216 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 520 T + p^{3} T^{2} \) |
| 53 | \( 1 + 146 T + p^{3} T^{2} \) |
| 59 | \( 1 + 460 T + p^{3} T^{2} \) |
| 61 | \( 1 - 628 T + p^{3} T^{2} \) |
| 67 | \( 1 + 556 T + p^{3} T^{2} \) |
| 71 | \( 1 + 592 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 79 | \( 1 - 104 T + p^{3} T^{2} \) |
| 83 | \( 1 - 324 T + p^{3} T^{2} \) |
| 89 | \( 1 - 896 T + p^{3} T^{2} \) |
| 97 | \( 1 + 920 T + p^{3} T^{2} \) |
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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
−8.239101053173136778159366724508, −7.60300640788790522980261092560, −6.69421703297279353925248868597, −5.98111052123501365137803298049, −5.06182788800019071263577386837, −4.04913167934432254321728817547, −3.36376918608296049431157971898, −2.20643555951314779646083608724, −1.48213819992926064015123183411, 0,
1.48213819992926064015123183411, 2.20643555951314779646083608724, 3.36376918608296049431157971898, 4.04913167934432254321728817547, 5.06182788800019071263577386837, 5.98111052123501365137803298049, 6.69421703297279353925248868597, 7.60300640788790522980261092560, 8.239101053173136778159366724508
