L(s) = 1 | − 5-s − 4·7-s − 4·11-s − 4·13-s − 4·17-s − 4·19-s − 8·23-s + 25-s + 6·29-s − 4·31-s + 4·35-s − 2·37-s − 10·41-s + 43-s − 4·47-s + 9·49-s − 2·53-s + 4·55-s − 12·59-s + 4·65-s − 8·67-s + 12·71-s − 14·73-s + 16·77-s + 8·79-s + 6·83-s + 4·85-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.51·7-s − 1.20·11-s − 1.10·13-s − 0.970·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s + 0.676·35-s − 0.328·37-s − 1.56·41-s + 0.152·43-s − 0.583·47-s + 9/7·49-s − 0.274·53-s + 0.539·55-s − 1.56·59-s + 0.496·65-s − 0.977·67-s + 1.42·71-s − 1.63·73-s + 1.82·77-s + 0.900·79-s + 0.658·83-s + 0.433·85-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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 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
−13.60965579971420, −13.31329721989767, −12.70740545190079, −12.40058217766289, −12.04296852554546, −11.40545861777014, −10.67826153881958, −10.34950938152807, −9.987900033382153, −9.483569357774897, −8.847375892781384, −8.417255213452447, −7.779877118573569, −7.412094682231946, −6.718997322993176, −6.370340196264069, −5.908237343438041, −5.055220178731097, −4.708856730659306, −4.052649737066978, −3.420666868552737, −2.904115656309310, −2.325927024397853, −1.804122056409562, −0.3705398502453019, 0,
0.3705398502453019, 1.804122056409562, 2.325927024397853, 2.904115656309310, 3.420666868552737, 4.052649737066978, 4.708856730659306, 5.055220178731097, 5.908237343438041, 6.370340196264069, 6.718997322993176, 7.412094682231946, 7.779877118573569, 8.417255213452447, 8.847375892781384, 9.483569357774897, 9.987900033382153, 10.34950938152807, 10.67826153881958, 11.40545861777014, 12.04296852554546, 12.40058217766289, 12.70740545190079, 13.31329721989767, 13.60965579971420