| L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s + 11-s − 6·13-s + 16-s − 17-s + 2·19-s + 20-s + 22-s + 2·23-s + 25-s − 6·26-s − 2·29-s + 4·31-s + 32-s − 34-s − 2·37-s + 2·38-s + 40-s − 12·41-s + 8·43-s + 44-s + 2·46-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s + 0.301·11-s − 1.66·13-s + 1/4·16-s − 0.242·17-s + 0.458·19-s + 0.223·20-s + 0.213·22-s + 0.417·23-s + 1/5·25-s − 1.17·26-s − 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.171·34-s − 0.328·37-s + 0.324·38-s + 0.158·40-s − 1.87·41-s + 1.21·43-s + 0.150·44-s + 0.294·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−16.13700138509635, −15.45457393494590, −14.91489013891292, −14.61062524222008, −13.77971303981305, −13.64780702401632, −12.82372989893896, −12.16796652167404, −12.07239410169262, −11.12063499018122, −10.72874763176906, −9.798392171771041, −9.632861048637097, −8.876762179498759, −7.943099165953826, −7.541768394854357, −6.632547860611924, −6.472982423547234, −5.417561054021447, −4.982295732040102, −4.492474493096454, −3.493943501979381, −2.887643150335628, −2.143773705051285, −1.370145308571343, 0,
1.370145308571343, 2.143773705051285, 2.887643150335628, 3.493943501979381, 4.492474493096454, 4.982295732040102, 5.417561054021447, 6.472982423547234, 6.632547860611924, 7.541768394854357, 7.943099165953826, 8.876762179498759, 9.632861048637097, 9.798392171771041, 10.72874763176906, 11.12063499018122, 12.07239410169262, 12.16796652167404, 12.82372989893896, 13.64780702401632, 13.77971303981305, 14.61062524222008, 14.91489013891292, 15.45457393494590, 16.13700138509635
