| L(s) = 1 | + 6·2-s − 4·3-s + 4·4-s − 24·6-s − 248·7-s − 168·8-s − 227·9-s + 204·11-s − 16·12-s + 370·13-s − 1.48e3·14-s − 1.13e3·16-s − 1.55e3·17-s − 1.36e3·18-s + 361·19-s + 992·21-s + 1.22e3·22-s + 408·23-s + 672·24-s + 2.22e3·26-s + 1.88e3·27-s − 992·28-s + 6.17e3·29-s − 7.84e3·31-s − 1.44e3·32-s − 816·33-s − 9.32e3·34-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 0.256·3-s + 1/8·4-s − 0.272·6-s − 1.91·7-s − 0.928·8-s − 0.934·9-s + 0.508·11-s − 0.0320·12-s + 0.607·13-s − 2.02·14-s − 1.10·16-s − 1.30·17-s − 0.990·18-s + 0.229·19-s + 0.490·21-s + 0.539·22-s + 0.160·23-s + 0.238·24-s + 0.644·26-s + 0.496·27-s − 0.239·28-s + 1.36·29-s − 1.46·31-s − 0.248·32-s − 0.130·33-s − 1.38·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.216686148\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216686148\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 - 3 p T + p^{5} T^{2} \) |
| 3 | \( 1 + 4 T + p^{5} T^{2} \) |
| 7 | \( 1 + 248 T + p^{5} T^{2} \) |
| 11 | \( 1 - 204 T + p^{5} T^{2} \) |
| 13 | \( 1 - 370 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1554 T + p^{5} T^{2} \) |
| 23 | \( 1 - 408 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6174 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7840 T + p^{5} T^{2} \) |
| 37 | \( 1 - 5146 T + p^{5} T^{2} \) |
| 41 | \( 1 + 7830 T + p^{5} T^{2} \) |
| 43 | \( 1 - 12532 T + p^{5} T^{2} \) |
| 47 | \( 1 + 2592 T + p^{5} T^{2} \) |
| 53 | \( 1 - 20778 T + p^{5} T^{2} \) |
| 59 | \( 1 - 18972 T + p^{5} T^{2} \) |
| 61 | \( 1 + 18418 T + p^{5} T^{2} \) |
| 67 | \( 1 - 11548 T + p^{5} T^{2} \) |
| 71 | \( 1 + 72984 T + p^{5} T^{2} \) |
| 73 | \( 1 + 59114 T + p^{5} T^{2} \) |
| 79 | \( 1 + 44752 T + p^{5} T^{2} \) |
| 83 | \( 1 - 27660 T + p^{5} T^{2} \) |
| 89 | \( 1 - 20730 T + p^{5} T^{2} \) |
| 97 | \( 1 + 14018 T + p^{5} T^{2} \) |
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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
−10.28077323712459094781750518298, −9.123793197707083738112009882840, −8.793650664815594220307920989085, −6.93371924354676711304446682576, −6.24362207016941858789845464728, −5.65387510399840208208289875990, −4.33425731484550035644425099549, −3.42678523471286582856818441028, −2.65814535980690305386206826010, −0.45771711034435106459869850727,
0.45771711034435106459869850727, 2.65814535980690305386206826010, 3.42678523471286582856818441028, 4.33425731484550035644425099549, 5.65387510399840208208289875990, 6.24362207016941858789845464728, 6.93371924354676711304446682576, 8.793650664815594220307920989085, 9.123793197707083738112009882840, 10.28077323712459094781750518298
