| L(s) = 1 | − 2·4-s + 5-s − 4·11-s + 4·16-s + 17-s + 4·19-s − 2·20-s + 8·23-s + 25-s + 5·29-s − 3·31-s − 2·37-s + 6·41-s − 10·43-s + 8·44-s + 10·47-s − 7·49-s − 5·53-s − 4·55-s + 3·59-s + 4·61-s − 8·64-s − 67-s − 2·68-s + 12·71-s + 8·73-s − 8·76-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 1.20·11-s + 16-s + 0.242·17-s + 0.917·19-s − 0.447·20-s + 1.66·23-s + 1/5·25-s + 0.928·29-s − 0.538·31-s − 0.328·37-s + 0.937·41-s − 1.52·43-s + 1.20·44-s + 1.45·47-s − 49-s − 0.686·53-s − 0.539·55-s + 0.390·59-s + 0.512·61-s − 64-s − 0.122·67-s − 0.242·68-s + 1.42·71-s + 0.936·73-s − 0.917·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 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
−13.64756214900577, −13.19298179474827, −12.98380384039734, −12.37184619156273, −12.00064827448595, −11.06103847675854, −10.93073792269133, −10.18548873022294, −9.874977135759820, −9.370002125502201, −8.912950457546741, −8.378065121171215, −7.910345557821361, −7.412169536504755, −6.823099216042143, −6.216731653839239, −5.409942849241444, −5.215038850502538, −4.869066137995989, −4.093223567350371, −3.407589091904047, −2.935488776780401, −2.370594656523685, −1.367609004533766, −0.8637543654156677, 0,
0.8637543654156677, 1.367609004533766, 2.370594656523685, 2.935488776780401, 3.407589091904047, 4.093223567350371, 4.869066137995989, 5.215038850502538, 5.409942849241444, 6.216731653839239, 6.823099216042143, 7.412169536504755, 7.910345557821361, 8.378065121171215, 8.912950457546741, 9.370002125502201, 9.874977135759820, 10.18548873022294, 10.93073792269133, 11.06103847675854, 12.00064827448595, 12.37184619156273, 12.98380384039734, 13.19298179474827, 13.64756214900577
