| L(s) = 1 | − 9·3-s + 694·7-s − 2.10e3·9-s − 4.90e3·11-s + 7.22e3·13-s + 1.59e4·17-s − 8.74e3·19-s − 6.24e3·21-s + 5.35e4·23-s + 3.86e4·27-s − 9.87e4·29-s + 6.50e4·31-s + 4.41e4·33-s − 3.23e5·37-s − 6.50e4·39-s + 1.70e5·41-s − 2.26e5·43-s − 3.99e5·47-s − 3.41e5·49-s − 1.43e5·51-s − 2.17e5·53-s + 7.87e4·57-s + 2.73e6·59-s − 1.41e6·61-s − 1.46e6·63-s + 3.16e6·67-s − 4.81e5·69-s + ⋯ |
| L(s) = 1 | − 0.192·3-s + 0.764·7-s − 0.962·9-s − 1.11·11-s + 0.912·13-s + 0.786·17-s − 0.292·19-s − 0.147·21-s + 0.917·23-s + 0.377·27-s − 0.751·29-s + 0.392·31-s + 0.213·33-s − 1.05·37-s − 0.175·39-s + 0.385·41-s − 0.433·43-s − 0.561·47-s − 0.415·49-s − 0.151·51-s − 0.200·53-s + 0.0563·57-s + 1.73·59-s − 0.795·61-s − 0.736·63-s + 1.28·67-s − 0.176·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + p^{2} T + p^{7} T^{2} \) |
| 7 | \( 1 - 694 T + p^{7} T^{2} \) |
| 11 | \( 1 + 4901 T + p^{7} T^{2} \) |
| 13 | \( 1 - 556 p T + p^{7} T^{2} \) |
| 17 | \( 1 - 15925 T + p^{7} T^{2} \) |
| 19 | \( 1 + 8749 T + p^{7} T^{2} \) |
| 23 | \( 1 - 53554 T + p^{7} T^{2} \) |
| 29 | \( 1 + 98752 T + p^{7} T^{2} \) |
| 31 | \( 1 - 65030 T + p^{7} T^{2} \) |
| 37 | \( 1 + 323958 T + p^{7} T^{2} \) |
| 41 | \( 1 - 170277 T + p^{7} T^{2} \) |
| 43 | \( 1 + 226228 T + p^{7} T^{2} \) |
| 47 | \( 1 + 399932 T + p^{7} T^{2} \) |
| 53 | \( 1 + 217218 T + p^{7} T^{2} \) |
| 59 | \( 1 - 2733128 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1410230 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3163693 T + p^{7} T^{2} \) |
| 71 | \( 1 - 3458532 T + p^{7} T^{2} \) |
| 73 | \( 1 - 4113051 T + p^{7} T^{2} \) |
| 79 | \( 1 - 34386 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1798039 T + p^{7} T^{2} \) |
| 89 | \( 1 + 12630321 T + p^{7} T^{2} \) |
| 97 | \( 1 + 7118942 T + p^{7} T^{2} \) |
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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
−9.671968498292332662695884128864, −8.453334610720774028243272755358, −8.054372260234894370758493095844, −6.79974973352156967630481102725, −5.59202270386274387092247489390, −5.06381457305019988792484793352, −3.61516824544293505750055547273, −2.53951378662225474046577087076, −1.24769623380995093591816435640, 0,
1.24769623380995093591816435640, 2.53951378662225474046577087076, 3.61516824544293505750055547273, 5.06381457305019988792484793352, 5.59202270386274387092247489390, 6.79974973352156967630481102725, 8.054372260234894370758493095844, 8.453334610720774028243272755358, 9.671968498292332662695884128864
