| L(s) = 1 | + 24·3-s − 172·7-s + 333·9-s − 132·11-s + 946·13-s + 222·17-s − 500·19-s − 4.12e3·21-s + 3.56e3·23-s + 2.16e3·27-s + 2.19e3·29-s − 2.31e3·31-s − 3.16e3·33-s + 1.12e4·37-s + 2.27e4·39-s + 1.24e3·41-s + 2.06e4·43-s + 6.58e3·47-s + 1.27e4·49-s + 5.32e3·51-s + 2.10e4·53-s − 1.20e4·57-s − 7.98e3·59-s + 1.66e4·61-s − 5.72e4·63-s + 1.80e3·67-s + 8.55e4·69-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 1.32·7-s + 1.37·9-s − 0.328·11-s + 1.55·13-s + 0.186·17-s − 0.317·19-s − 2.04·21-s + 1.40·23-s + 0.570·27-s + 0.483·29-s − 0.432·31-s − 0.506·33-s + 1.35·37-s + 2.39·39-s + 0.115·41-s + 1.70·43-s + 0.435·47-s + 0.760·49-s + 0.286·51-s + 1.03·53-s − 0.489·57-s − 0.298·59-s + 0.571·61-s − 1.81·63-s + 0.0492·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.554203552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.554203552\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 8 p T + p^{5} T^{2} \) |
| 7 | \( 1 + 172 T + p^{5} T^{2} \) |
| 11 | \( 1 + 12 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 946 T + p^{5} T^{2} \) |
| 17 | \( 1 - 222 T + p^{5} T^{2} \) |
| 19 | \( 1 + 500 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3564 T + p^{5} T^{2} \) |
| 29 | \( 1 - 2190 T + p^{5} T^{2} \) |
| 31 | \( 1 + 2312 T + p^{5} T^{2} \) |
| 37 | \( 1 - 11242 T + p^{5} T^{2} \) |
| 41 | \( 1 - 1242 T + p^{5} T^{2} \) |
| 43 | \( 1 - 20624 T + p^{5} T^{2} \) |
| 47 | \( 1 - 6588 T + p^{5} T^{2} \) |
| 53 | \( 1 - 21066 T + p^{5} T^{2} \) |
| 59 | \( 1 + 7980 T + p^{5} T^{2} \) |
| 61 | \( 1 - 16622 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1808 T + p^{5} T^{2} \) |
| 71 | \( 1 - 24528 T + p^{5} T^{2} \) |
| 73 | \( 1 + 20474 T + p^{5} T^{2} \) |
| 79 | \( 1 - 46240 T + p^{5} T^{2} \) |
| 83 | \( 1 + 51576 T + p^{5} T^{2} \) |
| 89 | \( 1 + 110310 T + p^{5} T^{2} \) |
| 97 | \( 1 - 78382 T + p^{5} T^{2} \) |
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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
−10.24394845784817453388211028322, −9.261729709051755189871456658286, −8.811659641743806043837545786100, −7.83455579511872992021731080829, −6.83672249537722280406326622905, −5.83496441988116811677371288147, −4.11038522984696904153830484632, −3.28146233130901227098852266694, −2.51402285945511922008059471115, −0.952756144615457139382659050003,
0.952756144615457139382659050003, 2.51402285945511922008059471115, 3.28146233130901227098852266694, 4.11038522984696904153830484632, 5.83496441988116811677371288147, 6.83672249537722280406326622905, 7.83455579511872992021731080829, 8.811659641743806043837545786100, 9.261729709051755189871456658286, 10.24394845784817453388211028322
