| L(s) = 1 | − 16·2-s + 46·3-s + 256·4-s − 625·5-s − 736·6-s − 1.03e4·7-s − 4.09e3·8-s − 1.75e4·9-s + 1.00e4·10-s − 5.56e3·11-s + 1.17e4·12-s + 4.59e4·13-s + 1.65e5·14-s − 2.87e4·15-s + 6.55e4·16-s − 3.81e5·17-s + 2.81e5·18-s + 6.10e5·19-s − 1.60e5·20-s − 4.74e5·21-s + 8.90e4·22-s − 1.44e6·23-s − 1.88e5·24-s + 3.90e5·25-s − 7.35e5·26-s − 1.71e6·27-s − 2.64e6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.327·3-s + 1/2·4-s − 0.447·5-s − 0.231·6-s − 1.62·7-s − 0.353·8-s − 0.892·9-s + 0.316·10-s − 0.114·11-s + 0.163·12-s + 0.446·13-s + 1.14·14-s − 0.146·15-s + 1/4·16-s − 1.10·17-s + 0.631·18-s + 1.07·19-s − 0.223·20-s − 0.532·21-s + 0.0810·22-s − 1.07·23-s − 0.115·24-s + 1/5·25-s − 0.315·26-s − 0.620·27-s − 0.812·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{4} T \) |
| 5 | \( 1 + p^{4} T \) |
| good | 3 | \( 1 - 46 T + p^{9} T^{2} \) |
| 7 | \( 1 + 1474 p T + p^{9} T^{2} \) |
| 11 | \( 1 + 5568 T + p^{9} T^{2} \) |
| 13 | \( 1 - 45986 T + p^{9} T^{2} \) |
| 17 | \( 1 + 381318 T + p^{9} T^{2} \) |
| 19 | \( 1 - 610460 T + p^{9} T^{2} \) |
| 23 | \( 1 + 1447914 T + p^{9} T^{2} \) |
| 29 | \( 1 - 5385510 T + p^{9} T^{2} \) |
| 31 | \( 1 - 3053852 T + p^{9} T^{2} \) |
| 37 | \( 1 - 12889442 T + p^{9} T^{2} \) |
| 41 | \( 1 + 33786618 T + p^{9} T^{2} \) |
| 43 | \( 1 + 36886234 T + p^{9} T^{2} \) |
| 47 | \( 1 + 44163798 T + p^{9} T^{2} \) |
| 53 | \( 1 - 29746266 T + p^{9} T^{2} \) |
| 59 | \( 1 + 65575380 T + p^{9} T^{2} \) |
| 61 | \( 1 - 40183202 T + p^{9} T^{2} \) |
| 67 | \( 1 + 115706158 T + p^{9} T^{2} \) |
| 71 | \( 1 + 231681708 T + p^{9} T^{2} \) |
| 73 | \( 1 - 358691906 T + p^{9} T^{2} \) |
| 79 | \( 1 + 486017080 T + p^{9} T^{2} \) |
| 83 | \( 1 - 251168886 T + p^{9} T^{2} \) |
| 89 | \( 1 + 526039110 T + p^{9} T^{2} \) |
| 97 | \( 1 + 1075981438 T + p^{9} T^{2} \) |
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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
−18.17674770234109574340073195289, −16.49760346673342694979299488176, −15.51869766232036129089545281682, −13.52884631526750982706488792743, −11.75007066646563119595358566331, −9.904625620639392787109259989547, −8.445358808011349762724740333691, −6.48036026027725390451528005228, −3.09665967626266841373478259759, 0,
3.09665967626266841373478259759, 6.48036026027725390451528005228, 8.445358808011349762724740333691, 9.904625620639392787109259989547, 11.75007066646563119595358566331, 13.52884631526750982706488792743, 15.51869766232036129089545281682, 16.49760346673342694979299488176, 18.17674770234109574340073195289
