L(s) = 1 | + 5-s + 4·11-s + 2·13-s + 2·17-s − 4·19-s − 8·23-s + 25-s − 6·29-s − 8·31-s − 2·37-s + 2·41-s + 12·43-s + 8·47-s − 6·53-s + 4·55-s − 4·59-s + 2·61-s + 2·65-s − 12·67-s + 8·71-s + 14·73-s − 12·83-s + 2·85-s + 2·89-s − 4·95-s − 10·97-s + 101-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.328·37-s + 0.312·41-s + 1.82·43-s + 1.16·47-s − 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s + 1.63·73-s − 1.31·83-s + 0.216·85-s + 0.211·89-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.16795006148422, −14.49722938803314, −14.18256531469436, −13.81528179061438, −13.03704333189100, −12.54184828900961, −12.16522546628426, −11.49307992936691, −10.83460493726009, −10.62451425894387, −9.693859274235424, −9.364293371310721, −8.880600999786153, −8.235924559134461, −7.584420305374129, −7.050631837309314, −6.263031867194771, −5.931433732237611, −5.443717927033889, −4.416452717023179, −3.925363061550828, −3.500111789336166, −2.397397754713740, −1.837860052810143, −1.142436955087004, 0,
1.142436955087004, 1.837860052810143, 2.397397754713740, 3.500111789336166, 3.925363061550828, 4.416452717023179, 5.443717927033889, 5.931433732237611, 6.263031867194771, 7.050631837309314, 7.584420305374129, 8.235924559134461, 8.880600999786153, 9.364293371310721, 9.693859274235424, 10.62451425894387, 10.83460493726009, 11.49307992936691, 12.16522546628426, 12.54184828900961, 13.03704333189100, 13.81528179061438, 14.18256531469436, 14.49722938803314, 15.16795006148422