| L(s) = 1 | − 2·3-s − 2·5-s + 7-s + 9-s − 4·11-s + 13-s + 4·15-s + 4·19-s − 2·21-s − 23-s − 25-s + 4·27-s + 6·29-s + 8·33-s − 2·35-s − 8·37-s − 2·39-s − 10·41-s + 12·43-s − 2·45-s − 12·47-s + 49-s − 6·53-s + 8·55-s − 8·57-s − 8·59-s + 63-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.03·15-s + 0.917·19-s − 0.436·21-s − 0.208·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s + 1.39·33-s − 0.338·35-s − 1.31·37-s − 0.320·39-s − 1.56·41-s + 1.82·43-s − 0.298·45-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 1.05·57-s − 1.04·59-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.02919969421498, −13.35034456398728, −12.89408421356825, −12.25453688613599, −11.94224938077956, −11.60419631080697, −11.14594635949726, −10.56748414276325, −10.34659109407842, −9.788786124445915, −8.947325406061818, −8.499932702167009, −8.016888847886506, −7.451018241807643, −7.214161861728100, −6.287577018647051, −6.062898125336093, −5.336083617260232, −4.888363567658581, −4.631795175440328, −3.796376805282935, −3.158621299040963, −2.718901243828256, −1.687824449387649, −1.095095921262981, 0, 0,
1.095095921262981, 1.687824449387649, 2.718901243828256, 3.158621299040963, 3.796376805282935, 4.631795175440328, 4.888363567658581, 5.336083617260232, 6.062898125336093, 6.287577018647051, 7.214161861728100, 7.451018241807643, 8.016888847886506, 8.499932702167009, 8.947325406061818, 9.788786124445915, 10.34659109407842, 10.56748414276325, 11.14594635949726, 11.60419631080697, 11.94224938077956, 12.25453688613599, 12.89408421356825, 13.35034456398728, 14.02919969421498
