| L(s) = 1 | + 3·3-s + 5·5-s + 20·7-s + 9·9-s + 24·11-s − 74·13-s + 15·15-s + 54·17-s + 124·19-s + 60·21-s − 120·23-s + 25·25-s + 27·27-s + 78·29-s + 200·31-s + 72·33-s + 100·35-s + 70·37-s − 222·39-s + 330·41-s − 92·43-s + 45·45-s − 24·47-s + 57·49-s + 162·51-s − 450·53-s + 120·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.07·7-s + 1/3·9-s + 0.657·11-s − 1.57·13-s + 0.258·15-s + 0.770·17-s + 1.49·19-s + 0.623·21-s − 1.08·23-s + 1/5·25-s + 0.192·27-s + 0.499·29-s + 1.15·31-s + 0.379·33-s + 0.482·35-s + 0.311·37-s − 0.911·39-s + 1.25·41-s − 0.326·43-s + 0.149·45-s − 0.0744·47-s + 0.166·49-s + 0.444·51-s − 1.16·53-s + 0.294·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.490636406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.490636406\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 - 20 T + p^{3} T^{2} \) |
| 11 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 74 T + p^{3} T^{2} \) |
| 17 | \( 1 - 54 T + p^{3} T^{2} \) |
| 19 | \( 1 - 124 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 - 78 T + p^{3} T^{2} \) |
| 31 | \( 1 - 200 T + p^{3} T^{2} \) |
| 37 | \( 1 - 70 T + p^{3} T^{2} \) |
| 41 | \( 1 - 330 T + p^{3} T^{2} \) |
| 43 | \( 1 + 92 T + p^{3} T^{2} \) |
| 47 | \( 1 + 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 450 T + p^{3} T^{2} \) |
| 59 | \( 1 + 24 T + p^{3} T^{2} \) |
| 61 | \( 1 - 322 T + p^{3} T^{2} \) |
| 67 | \( 1 - 196 T + p^{3} T^{2} \) |
| 71 | \( 1 + 288 T + p^{3} T^{2} \) |
| 73 | \( 1 + 430 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1026 T + p^{3} T^{2} \) |
| 97 | \( 1 + 286 T + p^{3} T^{2} \) |
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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
−9.820655707550078609761895223757, −8.827521793532370801591173782706, −7.82660707835892290418509931176, −7.43631824700695177940909957450, −6.19931163269611003231435590233, −5.13490386851902742457599975436, −4.43341997044823571914449870370, −3.12112864197034638428615951799, −2.10346259237287370778599960851, −1.05117822085005780212411993490,
1.05117822085005780212411993490, 2.10346259237287370778599960851, 3.12112864197034638428615951799, 4.43341997044823571914449870370, 5.13490386851902742457599975436, 6.19931163269611003231435590233, 7.43631824700695177940909957450, 7.82660707835892290418509931176, 8.827521793532370801591173782706, 9.820655707550078609761895223757
