| L(s) = 1 | + 9·3-s + 25·5-s + 80·7-s + 81·9-s − 684·11-s − 978·13-s + 225·15-s − 862·17-s − 916·19-s + 720·21-s + 1.55e3·23-s + 625·25-s + 729·27-s − 7.31e3·29-s + 9.31e3·31-s − 6.15e3·33-s + 2.00e3·35-s − 8.82e3·37-s − 8.80e3·39-s − 3.28e3·41-s − 7.55e3·43-s + 2.02e3·45-s + 5.96e3·47-s − 1.04e4·49-s − 7.75e3·51-s − 8.69e3·53-s − 1.71e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.617·7-s + 1/3·9-s − 1.70·11-s − 1.60·13-s + 0.258·15-s − 0.723·17-s − 0.582·19-s + 0.356·21-s + 0.611·23-s + 1/5·25-s + 0.192·27-s − 1.61·29-s + 1.74·31-s − 0.984·33-s + 0.275·35-s − 1.05·37-s − 0.926·39-s − 0.305·41-s − 0.623·43-s + 0.149·45-s + 0.393·47-s − 0.619·49-s − 0.417·51-s − 0.425·53-s − 0.762·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| good | 7 | \( 1 - 80 T + p^{5} T^{2} \) |
| 11 | \( 1 + 684 T + p^{5} T^{2} \) |
| 13 | \( 1 + 978 T + p^{5} T^{2} \) |
| 17 | \( 1 + 862 T + p^{5} T^{2} \) |
| 19 | \( 1 + 916 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1552 T + p^{5} T^{2} \) |
| 29 | \( 1 + 7314 T + p^{5} T^{2} \) |
| 31 | \( 1 - 9312 T + p^{5} T^{2} \) |
| 37 | \( 1 + 8826 T + p^{5} T^{2} \) |
| 41 | \( 1 + 3286 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7556 T + p^{5} T^{2} \) |
| 47 | \( 1 - 5960 T + p^{5} T^{2} \) |
| 53 | \( 1 + 8698 T + p^{5} T^{2} \) |
| 59 | \( 1 - 42036 T + p^{5} T^{2} \) |
| 61 | \( 1 - 37518 T + p^{5} T^{2} \) |
| 67 | \( 1 + 29324 T + p^{5} T^{2} \) |
| 71 | \( 1 + 84408 T + p^{5} T^{2} \) |
| 73 | \( 1 + 46550 T + p^{5} T^{2} \) |
| 79 | \( 1 + 26752 T + p^{5} T^{2} \) |
| 83 | \( 1 - 7956 T + p^{5} T^{2} \) |
| 89 | \( 1 - 59674 T + p^{5} T^{2} \) |
| 97 | \( 1 - 136898 T + p^{5} T^{2} \) |
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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
−10.56496334642201591743794611563, −9.915687847090816293931464757374, −8.752990508439255816404185403480, −7.84935974566042579472816702593, −6.97396418431101694581617572192, −5.37522389998955424217262032744, −4.60247625298195123214361585182, −2.80696315783005561482499047009, −1.99143793808855616200613721829, 0,
1.99143793808855616200613721829, 2.80696315783005561482499047009, 4.60247625298195123214361585182, 5.37522389998955424217262032744, 6.97396418431101694581617572192, 7.84935974566042579472816702593, 8.752990508439255816404185403480, 9.915687847090816293931464757374, 10.56496334642201591743794611563
