| L(s) = 1 | + 256·2-s + 4.62e3·3-s + 6.55e4·4-s − 8.51e5·5-s + 1.18e6·6-s + 5.76e6·7-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 + 1.47e9·14-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 + 2.66e10·21-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 + 3.77e11·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.407·3-s + 1/2·4-s − 0.975·5-s + 0.287·6-s + 0.377·7-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 + 0.267·14-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 + 0.153·21-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 + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{8} T \) |
| 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
−14.77902735687323229252001986844, −13.42937673378120912124336644627, −11.97426562821535849017684780304, −10.83680254819668421844796156003, −8.588114800693531103090252831258, −7.33237479472884021318513679667, −5.32520000022654732457145487690, −3.79854542067393623627951491960, −2.36888658027127046342327345811, 0,
2.36888658027127046342327345811, 3.79854542067393623627951491960, 5.32520000022654732457145487690, 7.33237479472884021318513679667, 8.588114800693531103090252831258, 10.83680254819668421844796156003, 11.97426562821535849017684780304, 13.42937673378120912124336644627, 14.77902735687323229252001986844
