| L(s) = 1 | − 5-s + 4·7-s + 2·13-s − 2·17-s − 4·19-s + 25-s + 2·29-s − 4·35-s − 10·37-s + 6·41-s + 43-s + 8·47-s + 9·49-s + 2·53-s − 2·61-s − 2·65-s + 4·67-s − 8·71-s + 10·73-s − 8·79-s + 16·83-s + 2·85-s − 6·89-s + 8·91-s + 4·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s − 0.676·35-s − 1.64·37-s + 0.937·41-s + 0.152·43-s + 1.16·47-s + 9/7·49-s + 0.274·53-s − 0.256·61-s − 0.248·65-s + 0.488·67-s − 0.949·71-s + 1.17·73-s − 0.900·79-s + 1.75·83-s + 0.216·85-s − 0.635·89-s + 0.838·91-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.89767898956441, −13.38173244472517, −12.71363907390979, −12.27598339487558, −11.82959454044330, −11.31278591438995, −10.83490469598836, −10.64621731061887, −10.03440528087575, −9.148841104930085, −8.779047802901478, −8.452305802613782, −7.823820004581707, −7.539296076114293, −6.795982444263490, −6.388912706235693, −5.635716891787019, −5.151354765872050, −4.634485108100675, −4.053944655665463, −3.743385300482935, −2.758108784022960, −2.215529886931365, −1.563639455226136, −0.9543272715444760, 0,
0.9543272715444760, 1.563639455226136, 2.215529886931365, 2.758108784022960, 3.743385300482935, 4.053944655665463, 4.634485108100675, 5.151354765872050, 5.635716891787019, 6.388912706235693, 6.795982444263490, 7.539296076114293, 7.823820004581707, 8.452305802613782, 8.779047802901478, 9.148841104930085, 10.03440528087575, 10.64621731061887, 10.83490469598836, 11.31278591438995, 11.82959454044330, 12.27598339487558, 12.71363907390979, 13.38173244472517, 13.89767898956441
