L(s) = 1 | − 5-s − 4·11-s − 6·13-s − 2·17-s + 4·19-s + 25-s + 6·29-s − 8·31-s + 6·37-s + 6·41-s − 8·43-s + 4·47-s + 2·53-s + 4·55-s − 10·61-s + 6·65-s + 8·67-s + 10·73-s − 4·79-s − 4·83-s + 2·85-s − 10·89-s − 4·95-s + 18·97-s + 101-s + 103-s + 107-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 1.20·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.986·37-s + 0.937·41-s − 1.21·43-s + 0.583·47-s + 0.274·53-s + 0.539·55-s − 1.28·61-s + 0.744·65-s + 0.977·67-s + 1.17·73-s − 0.450·79-s − 0.439·83-s + 0.216·85-s − 1.05·89-s − 0.410·95-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.36671466645762, −13.93406403753003, −13.35094757986194, −12.76118671948764, −12.44163783262949, −11.96278489994219, −11.29886917137870, −10.96918584807065, −10.24585579613563, −9.924607939168859, −9.330268716473735, −8.836923416657954, −7.951591808567473, −7.844505889088990, −7.184446223352429, −6.821973540201085, −5.954975129193989, −5.329461882922155, −4.934901231018025, −4.417045105401203, −3.684901579240581, −2.833885036672467, −2.600479783666773, −1.786857326917936, −0.7171044052122643, 0,
0.7171044052122643, 1.786857326917936, 2.600479783666773, 2.833885036672467, 3.684901579240581, 4.417045105401203, 4.934901231018025, 5.329461882922155, 5.954975129193989, 6.821973540201085, 7.184446223352429, 7.844505889088990, 7.951591808567473, 8.836923416657954, 9.330268716473735, 9.924607939168859, 10.24585579613563, 10.96918584807065, 11.29886917137870, 11.96278489994219, 12.44163783262949, 12.76118671948764, 13.35094757986194, 13.93406403753003, 14.36671466645762