| L(s) = 1 | + 27·3-s + 125·5-s − 416·7-s + 729·9-s + 3.94e3·11-s − 7.97e3·13-s + 3.37e3·15-s − 3.40e4·17-s − 3.02e3·19-s − 1.12e4·21-s + 6.65e4·23-s + 1.56e4·25-s + 1.96e4·27-s − 1.85e4·29-s − 2.08e5·31-s + 1.06e5·33-s − 5.20e4·35-s − 3.01e5·37-s − 2.15e5·39-s − 4.60e5·41-s − 3.43e5·43-s + 9.11e4·45-s + 1.35e6·47-s − 6.50e5·49-s − 9.18e5·51-s + 6.17e5·53-s + 4.93e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.458·7-s + 1/3·9-s + 0.894·11-s − 1.00·13-s + 0.258·15-s − 1.67·17-s − 0.101·19-s − 0.264·21-s + 1.14·23-s + 1/5·25-s + 0.192·27-s − 0.141·29-s − 1.25·31-s + 0.516·33-s − 0.205·35-s − 0.978·37-s − 0.581·39-s − 1.04·41-s − 0.657·43-s + 0.149·45-s + 1.90·47-s − 0.789·49-s − 0.969·51-s + 0.570·53-s + 0.399·55-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 | \( 1 + 416 T + p^{7} T^{2} \) |
| 11 | \( 1 - 3948 T + p^{7} T^{2} \) |
| 13 | \( 1 + 7978 T + p^{7} T^{2} \) |
| 17 | \( 1 + 34014 T + p^{7} T^{2} \) |
| 19 | \( 1 + 3020 T + p^{7} T^{2} \) |
| 23 | \( 1 - 66528 T + p^{7} T^{2} \) |
| 29 | \( 1 + 18570 T + p^{7} T^{2} \) |
| 31 | \( 1 + 208832 T + p^{7} T^{2} \) |
| 37 | \( 1 + 301474 T + p^{7} T^{2} \) |
| 41 | \( 1 + 460998 T + p^{7} T^{2} \) |
| 43 | \( 1 + 343052 T + p^{7} T^{2} \) |
| 47 | \( 1 - 1356264 T + p^{7} T^{2} \) |
| 53 | \( 1 - 617982 T + p^{7} T^{2} \) |
| 59 | \( 1 + 939540 T + p^{7} T^{2} \) |
| 61 | \( 1 + 204298 T + p^{7} T^{2} \) |
| 67 | \( 1 - 758524 T + p^{7} T^{2} \) |
| 71 | \( 1 + 912072 T + p^{7} T^{2} \) |
| 73 | \( 1 - 3043322 T + p^{7} T^{2} \) |
| 79 | \( 1 + 6010880 T + p^{7} T^{2} \) |
| 83 | \( 1 + 9723012 T + p^{7} T^{2} \) |
| 89 | \( 1 - 7160010 T + p^{7} T^{2} \) |
| 97 | \( 1 + 16785214 T + p^{7} T^{2} \) |
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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
−10.25378524322437556988906130248, −9.233991110095345750611401719489, −8.776287541064830686266184400127, −7.21517727664568081365631318739, −6.59395685601277405521522941043, −5.15229165582765296830793153963, −3.97113635388471876299734368909, −2.71774044830319255384290856182, −1.65301422017524036799211023571, 0,
1.65301422017524036799211023571, 2.71774044830319255384290856182, 3.97113635388471876299734368909, 5.15229165582765296830793153963, 6.59395685601277405521522941043, 7.21517727664568081365631318739, 8.776287541064830686266184400127, 9.233991110095345750611401719489, 10.25378524322437556988906130248
