| L(s) = 1 | + 27·3-s + 125·5-s + 832·7-s + 729·9-s − 3.15e3·11-s − 7.69e3·13-s + 3.37e3·15-s + 258·17-s − 4.57e4·19-s + 2.24e4·21-s − 1.04e5·23-s + 1.56e4·25-s + 1.96e4·27-s + 3.86e4·29-s − 1.92e5·31-s − 8.52e4·33-s + 1.04e5·35-s + 4.03e5·37-s − 2.07e5·39-s + 8.60e4·41-s + 1.27e5·43-s + 9.11e4·45-s − 6.01e5·47-s − 1.31e5·49-s + 6.96e3·51-s − 1.62e6·53-s − 3.94e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.916·7-s + 1/3·9-s − 0.714·11-s − 0.970·13-s + 0.258·15-s + 0.0127·17-s − 1.52·19-s + 0.529·21-s − 1.79·23-s + 1/5·25-s + 0.192·27-s + 0.294·29-s − 1.15·31-s − 0.412·33-s + 0.410·35-s + 1.30·37-s − 0.560·39-s + 0.194·41-s + 0.244·43-s + 0.149·45-s − 0.844·47-s − 0.159·49-s + 0.00735·51-s − 1.50·53-s − 0.319·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p^{3} T \) |
| 5 | \( 1 - p^{3} T \) |
| good | 7 | \( 1 - 832 T + p^{7} T^{2} \) |
| 11 | \( 1 + 3156 T + p^{7} T^{2} \) |
| 13 | \( 1 + 7690 T + p^{7} T^{2} \) |
| 17 | \( 1 - 258 T + p^{7} T^{2} \) |
| 19 | \( 1 + 45740 T + p^{7} T^{2} \) |
| 23 | \( 1 + 104832 T + p^{7} T^{2} \) |
| 29 | \( 1 - 38646 T + p^{7} T^{2} \) |
| 31 | \( 1 + 192224 T + p^{7} T^{2} \) |
| 37 | \( 1 - 403454 T + p^{7} T^{2} \) |
| 41 | \( 1 - 86010 T + p^{7} T^{2} \) |
| 43 | \( 1 - 127348 T + p^{7} T^{2} \) |
| 47 | \( 1 + 601272 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1628226 T + p^{7} T^{2} \) |
| 59 | \( 1 + 198996 T + p^{7} T^{2} \) |
| 61 | \( 1 - 1209782 T + p^{7} T^{2} \) |
| 67 | \( 1 - 699388 T + p^{7} T^{2} \) |
| 71 | \( 1 - 4939320 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1275334 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6559712 T + p^{7} T^{2} \) |
| 83 | \( 1 - 3108348 T + p^{7} T^{2} \) |
| 89 | \( 1 - 5542410 T + p^{7} T^{2} \) |
| 97 | \( 1 - 4513346 T + p^{7} T^{2} \) |
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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
−10.30837360962139738453507454596, −9.467347034268125317110871856247, −8.265424095586554527399835703100, −7.70838416614435932154731349149, −6.34854527126367288150948080787, −5.11488013084283388447265434761, −4.13315647707977240214204607538, −2.51524909791404662657692278835, −1.78339885647917103745990206487, 0,
1.78339885647917103745990206487, 2.51524909791404662657692278835, 4.13315647707977240214204607538, 5.11488013084283388447265434761, 6.34854527126367288150948080787, 7.70838416614435932154731349149, 8.265424095586554527399835703100, 9.467347034268125317110871856247, 10.30837360962139738453507454596
