| L(s) = 1 | + 0.732·3-s − 5-s + 3.46·7-s − 2.46·9-s + 2.73·11-s + 13-s − 0.732·15-s + 7.46·17-s − 2.73·19-s + 2.53·21-s − 0.732·23-s + 25-s − 4·27-s + 9.46·29-s − 0.196·31-s + 2·33-s − 3.46·35-s + 0.732·39-s + 0.535·41-s − 7.26·43-s + 2.46·45-s + 4.53·47-s + 4.99·49-s + 5.46·51-s − 0.535·53-s − 2.73·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.422·3-s − 0.447·5-s + 1.30·7-s − 0.821·9-s + 0.823·11-s + 0.277·13-s − 0.189·15-s + 1.81·17-s − 0.626·19-s + 0.553·21-s − 0.152·23-s + 0.200·25-s − 0.769·27-s + 1.75·29-s − 0.0352·31-s + 0.348·33-s − 0.585·35-s + 0.117·39-s + 0.0836·41-s − 1.10·43-s + 0.367·45-s + 0.661·47-s + 0.714·49-s + 0.765·51-s − 0.0736·53-s − 0.368·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.763002430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.763002430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 17 | \( 1 - 7.46T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 + 0.732T + 23T^{2} \) |
| 29 | \( 1 - 9.46T + 29T^{2} \) |
| 31 | \( 1 + 0.196T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 0.535T + 41T^{2} \) |
| 43 | \( 1 + 7.26T + 43T^{2} \) |
| 47 | \( 1 - 4.53T + 47T^{2} \) |
| 53 | \( 1 + 0.535T + 53T^{2} \) |
| 59 | \( 1 - 5.66T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 9.26T + 71T^{2} \) |
| 73 | \( 1 + 2.92T + 73T^{2} \) |
| 79 | \( 1 + 6.53T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 15.8T + 97T^{2} \) |
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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
−10.97600575780112934316907505504, −10.02924581371377934703209868279, −8.793363143840068806807251551510, −8.274658006708178020450122064181, −7.53844637418609047564072990656, −6.23744849259013932365528684669, −5.16920999808620702659828679307, −4.09043065945765163757391542059, −2.95818306668519194005910648112, −1.38230213712586315015274173446,
1.38230213712586315015274173446, 2.95818306668519194005910648112, 4.09043065945765163757391542059, 5.16920999808620702659828679307, 6.23744849259013932365528684669, 7.53844637418609047564072990656, 8.274658006708178020450122064181, 8.793363143840068806807251551510, 10.02924581371377934703209868279, 10.97600575780112934316907505504
