L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 7-s + 2·10-s + 5·11-s − 13-s − 2·14-s − 4·16-s − 6·19-s − 2·20-s − 10·22-s + 3·23-s + 25-s + 2·26-s + 2·28-s + 2·29-s − 2·31-s + 8·32-s − 35-s − 7·37-s + 12·38-s + 9·41-s − 8·43-s + 10·44-s − 6·46-s − 10·47-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.377·7-s + 0.632·10-s + 1.50·11-s − 0.277·13-s − 0.534·14-s − 16-s − 1.37·19-s − 0.447·20-s − 2.13·22-s + 0.625·23-s + 1/5·25-s + 0.392·26-s + 0.377·28-s + 0.371·29-s − 0.359·31-s + 1.41·32-s − 0.169·35-s − 1.15·37-s + 1.94·38-s + 1.40·41-s − 1.21·43-s + 1.50·44-s − 0.884·46-s − 1.45·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5403458489\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5403458489\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−13.19443304543441, −12.42672648453859, −12.34803790340648, −11.47407121123791, −11.25288535011090, −10.90466547092126, −10.25200702985302, −9.854698474447139, −9.233638816995945, −8.971474856835731, −8.484977284042933, −8.043718447818127, −7.609088112506001, −6.971714004308585, −6.467305374906319, −6.368551656927016, −5.212918875336158, −4.779232439569671, −4.211802636451167, −3.700723360249858, −2.996771924523271, −2.172231173706090, −1.608686666761158, −1.180504422641399, −0.2913037367247719,
0.2913037367247719, 1.180504422641399, 1.608686666761158, 2.172231173706090, 2.996771924523271, 3.700723360249858, 4.211802636451167, 4.779232439569671, 5.212918875336158, 6.368551656927016, 6.467305374906319, 6.971714004308585, 7.609088112506001, 8.043718447818127, 8.484977284042933, 8.971474856835731, 9.233638816995945, 9.854698474447139, 10.25200702985302, 10.90466547092126, 11.25288535011090, 11.47407121123791, 12.34803790340648, 12.42672648453859, 13.19443304543441