| L(s) = 1 | − 5-s + 4·7-s + 4·11-s − 6·17-s − 19-s − 8·23-s + 25-s + 6·29-s − 8·31-s − 4·35-s − 8·37-s + 2·41-s − 12·47-s + 9·49-s − 4·53-s − 4·55-s − 8·59-s − 14·61-s − 2·67-s + 8·71-s − 2·73-s + 16·77-s + 4·79-s − 12·83-s + 6·85-s − 6·89-s + 95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s + 1.20·11-s − 1.45·17-s − 0.229·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.676·35-s − 1.31·37-s + 0.312·41-s − 1.75·47-s + 9/7·49-s − 0.549·53-s − 0.539·55-s − 1.04·59-s − 1.79·61-s − 0.244·67-s + 0.949·71-s − 0.234·73-s + 1.82·77-s + 0.450·79-s − 1.31·83-s + 0.650·85-s − 0.635·89-s + 0.102·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.76369370231339701324152191839, −6.88041329314991313162908747267, −6.34967601647381901827378106493, −5.41667675768737194603825986643, −4.47425321394436464061194531460, −4.28293978814797362391843511784, −3.27994314782227103907635898308, −1.97319590905619049254726094480, −1.53954673500509334685976132001, 0,
1.53954673500509334685976132001, 1.97319590905619049254726094480, 3.27994314782227103907635898308, 4.28293978814797362391843511784, 4.47425321394436464061194531460, 5.41667675768737194603825986643, 6.34967601647381901827378106493, 6.88041329314991313162908747267, 7.76369370231339701324152191839
