| L(s) = 1 | − 5-s − 4·11-s + 4·13-s + 4·17-s − 4·19-s − 23-s + 25-s + 6·29-s − 4·31-s − 2·37-s + 4·43-s − 7·49-s + 6·53-s + 4·55-s − 6·59-s − 4·61-s − 4·65-s − 4·67-s − 4·71-s − 2·73-s − 10·79-s + 18·83-s − 4·85-s + 2·89-s + 4·95-s + 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s + 1.10·13-s + 0.970·17-s − 0.917·19-s − 0.208·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.609·43-s − 49-s + 0.824·53-s + 0.539·55-s − 0.781·59-s − 0.512·61-s − 0.496·65-s − 0.488·67-s − 0.474·71-s − 0.234·73-s − 1.12·79-s + 1.97·83-s − 0.433·85-s + 0.211·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−16.12370789009967, −15.79550959186122, −15.03716847346628, −14.70981527289129, −13.84420258186783, −13.51279978783403, −12.76373697740170, −12.41762460337405, −11.75157014861805, −11.05135687567463, −10.57305810677622, −10.20636662948828, −9.355078328944954, −8.638874730490314, −8.195408537148375, −7.674426618210914, −7.018784773207035, −6.186854119481533, −5.733945627034597, −4.940458725158175, −4.310878056208900, −3.504186660123175, −2.951325017536565, −2.031711320473968, −1.063067370222496, 0,
1.063067370222496, 2.031711320473968, 2.951325017536565, 3.504186660123175, 4.310878056208900, 4.940458725158175, 5.733945627034597, 6.186854119481533, 7.018784773207035, 7.674426618210914, 8.195408537148375, 8.638874730490314, 9.355078328944954, 10.20636662948828, 10.57305810677622, 11.05135687567463, 11.75157014861805, 12.41762460337405, 12.76373697740170, 13.51279978783403, 13.84420258186783, 14.70981527289129, 15.03716847346628, 15.79550959186122, 16.12370789009967
