| L(s) = 1 | − 27·3-s − 125·5-s + 420·7-s + 729·9-s + 2.94e3·11-s − 1.10e4·13-s + 3.37e3·15-s − 1.65e4·17-s + 2.53e4·19-s − 1.13e4·21-s + 5.88e3·23-s + 1.56e4·25-s − 1.96e4·27-s + 1.63e5·29-s + 2.01e5·31-s − 7.94e4·33-s − 5.25e4·35-s + 1.20e5·37-s + 2.97e5·39-s − 1.15e5·41-s + 1.91e4·43-s − 9.11e4·45-s − 8.41e5·47-s − 6.47e5·49-s + 4.46e5·51-s + 5.01e5·53-s − 3.68e5·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.462·7-s + 1/3·9-s + 0.666·11-s − 1.38·13-s + 0.258·15-s − 0.816·17-s + 0.848·19-s − 0.267·21-s + 0.100·23-s + 1/5·25-s − 0.192·27-s + 1.24·29-s + 1.21·31-s − 0.385·33-s − 0.206·35-s + 0.391·37-s + 0.802·39-s − 0.262·41-s + 0.0367·43-s − 0.149·45-s − 1.18·47-s − 0.785·49-s + 0.471·51-s + 0.463·53-s − 0.298·55-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 | \( 1 - 60 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 2944 T + p^{7} T^{2} \) |
| 13 | \( 1 + 11006 T + p^{7} T^{2} \) |
| 17 | \( 1 + 16546 T + p^{7} T^{2} \) |
| 19 | \( 1 - 25364 T + p^{7} T^{2} \) |
| 23 | \( 1 - 5880 T + p^{7} T^{2} \) |
| 29 | \( 1 - 163042 T + p^{7} T^{2} \) |
| 31 | \( 1 - 201600 T + p^{7} T^{2} \) |
| 37 | \( 1 - 120530 T + p^{7} T^{2} \) |
| 41 | \( 1 + 115910 T + p^{7} T^{2} \) |
| 43 | \( 1 - 19148 T + p^{7} T^{2} \) |
| 47 | \( 1 + 841016 T + p^{7} T^{2} \) |
| 53 | \( 1 - 501890 T + p^{7} T^{2} \) |
| 59 | \( 1 - 1586176 T + p^{7} T^{2} \) |
| 61 | \( 1 + 372962 T + p^{7} T^{2} \) |
| 67 | \( 1 + 4561044 T + p^{7} T^{2} \) |
| 71 | \( 1 + 1512832 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1522910 T + p^{7} T^{2} \) |
| 79 | \( 1 + 4231920 T + p^{7} T^{2} \) |
| 83 | \( 1 - 1854204 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6888174 T + p^{7} T^{2} \) |
| 97 | \( 1 - 3700034 T + p^{7} T^{2} \) |
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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
−10.43679087070905434079884618569, −9.544946286913014817199306245510, −8.359162232060906554728761679388, −7.31184987585715540805236131889, −6.44070595755268643263406509455, −5.04699293808199293958660310710, −4.33689058749988396473099477097, −2.77188412474632822976651503285, −1.26857795096835826408736903492, 0,
1.26857795096835826408736903492, 2.77188412474632822976651503285, 4.33689058749988396473099477097, 5.04699293808199293958660310710, 6.44070595755268643263406509455, 7.31184987585715540805236131889, 8.359162232060906554728761679388, 9.544946286913014817199306245510, 10.43679087070905434079884618569
