L(s) = 1 | + 5-s − 4·7-s + 4·11-s + 2·13-s + 2·17-s − 19-s − 8·23-s + 25-s + 6·29-s − 4·31-s − 4·35-s + 10·37-s + 2·41-s + 12·43-s + 9·49-s + 6·53-s + 4·55-s + 10·61-s + 2·65-s − 4·67-s − 8·71-s + 2·73-s − 16·77-s + 12·79-s + 8·83-s + 2·85-s − 6·89-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.229·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.676·35-s + 1.64·37-s + 0.312·41-s + 1.82·43-s + 9/7·49-s + 0.824·53-s + 0.539·55-s + 1.28·61-s + 0.248·65-s − 0.488·67-s − 0.949·71-s + 0.234·73-s − 1.82·77-s + 1.35·79-s + 0.878·83-s + 0.216·85-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.450304025\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.450304025\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.37601394554457, −13.90022607834141, −13.41303243706700, −12.88555744810643, −12.40577559159163, −11.98659840942267, −11.43944750797710, −10.67936185777349, −10.26089886561285, −9.656448977317826, −9.371156550688042, −8.873985633158084, −8.191321424393345, −7.566594324387630, −6.868643450488899, −6.357149844663051, −6.011639763176624, −5.624598796302091, −4.547544650399926, −3.910788648885748, −3.636526887390439, −2.745939320124750, −2.230656685411676, −1.236959620494122, −0.5903830058358968,
0.5903830058358968, 1.236959620494122, 2.230656685411676, 2.745939320124750, 3.636526887390439, 3.910788648885748, 4.547544650399926, 5.624598796302091, 6.011639763176624, 6.357149844663051, 6.868643450488899, 7.566594324387630, 8.191321424393345, 8.873985633158084, 9.371156550688042, 9.656448977317826, 10.26089886561285, 10.67936185777349, 11.43944750797710, 11.98659840942267, 12.40577559159163, 12.88555744810643, 13.41303243706700, 13.90022607834141, 14.37601394554457