| L(s) = 1 | − 16·2-s − 81·3-s + 256·4-s − 625·5-s + 1.29e3·6-s + 6.33e3·7-s − 4.09e3·8-s + 6.56e3·9-s + 1.00e4·10-s + 7.75e3·11-s − 2.07e4·12-s + 7.26e4·13-s − 1.01e5·14-s + 5.06e4·15-s + 6.55e4·16-s − 3.34e5·17-s − 1.04e5·18-s − 9.34e5·19-s − 1.60e5·20-s − 5.12e5·21-s − 1.24e5·22-s − 9.91e5·23-s + 3.31e5·24-s + 3.90e5·25-s − 1.16e6·26-s − 5.31e5·27-s + 1.62e6·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.996·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.159·11-s − 0.288·12-s + 0.705·13-s − 0.704·14-s + 0.258·15-s + 1/4·16-s − 0.971·17-s − 0.235·18-s − 1.64·19-s − 0.223·20-s − 0.575·21-s − 0.112·22-s − 0.738·23-s + 0.204·24-s + 1/5·25-s − 0.498·26-s − 0.192·27-s + 0.498·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{4} T \) |
| 3 | \( 1 + p^{4} T \) |
| 5 | \( 1 + p^{4} T \) |
| good | 7 | \( 1 - 6332 T + p^{9} T^{2} \) |
| 11 | \( 1 - 7752 T + p^{9} T^{2} \) |
| 13 | \( 1 - 72626 T + p^{9} T^{2} \) |
| 17 | \( 1 + 334698 T + p^{9} T^{2} \) |
| 19 | \( 1 + 934660 T + p^{9} T^{2} \) |
| 23 | \( 1 + 991704 T + p^{9} T^{2} \) |
| 29 | \( 1 + 3638790 T + p^{9} T^{2} \) |
| 31 | \( 1 + 6063688 T + p^{9} T^{2} \) |
| 37 | \( 1 - 12489842 T + p^{9} T^{2} \) |
| 41 | \( 1 + 5035398 T + p^{9} T^{2} \) |
| 43 | \( 1 - 30163316 T + p^{9} T^{2} \) |
| 47 | \( 1 + 743928 T + p^{9} T^{2} \) |
| 53 | \( 1 + 102388134 T + p^{9} T^{2} \) |
| 59 | \( 1 + 49464840 T + p^{9} T^{2} \) |
| 61 | \( 1 + 130545898 T + p^{9} T^{2} \) |
| 67 | \( 1 - 102905012 T + p^{9} T^{2} \) |
| 71 | \( 1 + 190423008 T + p^{9} T^{2} \) |
| 73 | \( 1 + 367621054 T + p^{9} T^{2} \) |
| 79 | \( 1 - 175880360 T + p^{9} T^{2} \) |
| 83 | \( 1 + 100482444 T + p^{9} T^{2} \) |
| 89 | \( 1 + 660904110 T + p^{9} T^{2} \) |
| 97 | \( 1 - 1321991522 T + p^{9} T^{2} \) |
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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
−14.63330731020284037645072392855, −12.82679630404834726985780276107, −11.37483984003614993070821900736, −10.77979357808863839846644826561, −8.945193643532061105983837905685, −7.73388401013009046033064927534, −6.18356425069232485776447133116, −4.30424035883560361722790727801, −1.76411988055228078121422831521, 0,
1.76411988055228078121422831521, 4.30424035883560361722790727801, 6.18356425069232485776447133116, 7.73388401013009046033064927534, 8.945193643532061105983837905685, 10.77979357808863839846644826561, 11.37483984003614993070821900736, 12.82679630404834726985780276107, 14.63330731020284037645072392855
