| L(s) = 1 | + 256·2-s − 4.62e3·3-s + 6.55e4·4-s + 8.51e5·5-s − 1.18e6·6-s + 1.67e7·8-s − 1.07e8·9-s + 2.18e8·10-s − 5.86e8·11-s − 3.03e8·12-s + 1.04e9·13-s − 3.93e9·15-s + 4.29e9·16-s + 1.71e10·17-s − 2.75e10·18-s + 3.52e10·19-s + 5.58e10·20-s − 1.50e11·22-s + 2.26e11·23-s − 7.76e10·24-s − 3.75e10·25-s + 2.66e11·26-s + 1.09e12·27-s − 3.38e12·29-s − 1.00e12·30-s − 2.57e11·31-s + 1.09e12·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.407·3-s + 1/2·4-s + 0.975·5-s − 0.287·6-s + 0.353·8-s − 0.834·9-s + 0.689·10-s − 0.824·11-s − 0.203·12-s + 0.354·13-s − 0.396·15-s + 1/4·16-s + 0.597·17-s − 0.589·18-s + 0.476·19-s + 0.487·20-s − 0.582·22-s + 0.602·23-s − 0.143·24-s − 0.0492·25-s + 0.250·26-s + 0.746·27-s − 1.25·29-s − 0.280·30-s − 0.0541·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{8} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 514 p^{2} T + p^{17} T^{2} \) |
| 5 | \( 1 - 34068 p^{2} T + p^{17} T^{2} \) |
| 11 | \( 1 + 586048992 T + p^{17} T^{2} \) |
| 13 | \( 1 - 80228176 p T + p^{17} T^{2} \) |
| 17 | \( 1 - 17187488802 T + p^{17} T^{2} \) |
| 19 | \( 1 - 35251814482 T + p^{17} T^{2} \) |
| 23 | \( 1 - 226463988840 T + p^{17} T^{2} \) |
| 29 | \( 1 + 3381208637406 T + p^{17} T^{2} \) |
| 31 | \( 1 + 257228086436 T + p^{17} T^{2} \) |
| 37 | \( 1 + 40457204426662 T + p^{17} T^{2} \) |
| 41 | \( 1 - 29013168626274 T + p^{17} T^{2} \) |
| 43 | \( 1 - 12667778737448 T + p^{17} T^{2} \) |
| 47 | \( 1 + 129865524636 p^{2} T + p^{17} T^{2} \) |
| 53 | \( 1 - 564480420537078 T + p^{17} T^{2} \) |
| 59 | \( 1 + 1802377718625462 T + p^{17} T^{2} \) |
| 61 | \( 1 - 668064962693740 T + p^{17} T^{2} \) |
| 67 | \( 1 - 332890586370548 T + p^{17} T^{2} \) |
| 71 | \( 1 - 4451829225077376 T + p^{17} T^{2} \) |
| 73 | \( 1 - 6135974687950990 T + p^{17} T^{2} \) |
| 79 | \( 1 - 778901092563704 T + p^{17} T^{2} \) |
| 83 | \( 1 - 107044950027918 p T + p^{17} T^{2} \) |
| 89 | \( 1 + 29468733723774090 T + p^{17} T^{2} \) |
| 97 | \( 1 - 44579205354290530 T + p^{17} T^{2} \) |
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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
−10.42702102235848865565080044578, −9.284716545841766108043099857974, −7.952095759680970815311611295945, −6.65014560603001494498626344786, −5.56121838006146036865303182491, −5.21908366580235402835732477870, −3.52781568851982432087395531805, −2.52386294958797553512849327192, −1.42023404320432685034069717888, 0,
1.42023404320432685034069717888, 2.52386294958797553512849327192, 3.52781568851982432087395531805, 5.21908366580235402835732477870, 5.56121838006146036865303182491, 6.65014560603001494498626344786, 7.952095759680970815311611295945, 9.284716545841766108043099857974, 10.42702102235848865565080044578
