| L(s) = 1 | − 3-s + 1.41·7-s + 9-s + 1.41·13-s − 1.41·21-s + 25-s − 27-s − 1.41·31-s − 1.41·37-s − 1.41·39-s + 1.00·49-s + 1.41·61-s + 1.41·63-s + 2·67-s − 75-s − 1.41·79-s + 81-s + 2.00·91-s + 1.41·93-s − 1.41·103-s − 1.41·109-s + 1.41·111-s + 1.41·117-s + ⋯ |
| L(s) = 1 | − 3-s + 1.41·7-s + 9-s + 1.41·13-s − 1.41·21-s + 25-s − 27-s − 1.41·31-s − 1.41·37-s − 1.41·39-s + 1.00·49-s + 1.41·61-s + 1.41·63-s + 2·67-s − 75-s − 1.41·79-s + 81-s + 2.00·91-s + 1.41·93-s − 1.41·103-s − 1.41·109-s + 1.41·111-s + 1.41·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.163767079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163767079\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 - 1.41T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 1.41T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 + 1.41T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 1.41T + T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + 1.41T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.699632468882944135308092720186, −8.238381968969571126504455675225, −7.24435152408793876969587731719, −6.66710307341256271006750295899, −5.63281229772503130206451191561, −5.21914527619629865995098436437, −4.33659027224392614150832295957, −3.54574700704200518012477746876, −1.94112377061757138325208870068, −1.13500180564826304102144554184,
1.13500180564826304102144554184, 1.94112377061757138325208870068, 3.54574700704200518012477746876, 4.33659027224392614150832295957, 5.21914527619629865995098436437, 5.63281229772503130206451191561, 6.66710307341256271006750295899, 7.24435152408793876969587731719, 8.238381968969571126504455675225, 8.699632468882944135308092720186
