| L(s) = 1 | − 36·2-s + 81·3-s + 784·4-s + 1.31e3·5-s − 2.91e3·6-s − 9.79e3·8-s + 6.56e3·9-s − 4.73e4·10-s + 1.47e3·11-s + 6.35e4·12-s + 1.51e5·13-s + 1.06e5·15-s − 4.88e4·16-s − 1.08e5·17-s − 2.36e5·18-s − 5.93e5·19-s + 1.03e6·20-s − 5.31e4·22-s − 9.69e5·23-s − 7.93e5·24-s − 2.26e5·25-s − 5.45e6·26-s + 5.31e5·27-s − 6.64e6·29-s − 3.83e6·30-s − 7.07e6·31-s + 6.77e6·32-s + ⋯ |
| L(s) = 1 | − 1.59·2-s + 0.577·3-s + 1.53·4-s + 0.940·5-s − 0.918·6-s − 0.845·8-s + 1/3·9-s − 1.49·10-s + 0.0303·11-s + 0.884·12-s + 1.47·13-s + 0.542·15-s − 0.186·16-s − 0.314·17-s − 0.530·18-s − 1.04·19-s + 1.43·20-s − 0.0483·22-s − 0.722·23-s − 0.487·24-s − 0.115·25-s − 2.34·26-s + 0.192·27-s − 1.74·29-s − 0.863·30-s − 1.37·31-s + 1.14·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p^{4} T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 9 p^{2} T + p^{9} T^{2} \) |
| 5 | \( 1 - 1314 T + p^{9} T^{2} \) |
| 11 | \( 1 - 1476 T + p^{9} T^{2} \) |
| 13 | \( 1 - 151522 T + p^{9} T^{2} \) |
| 17 | \( 1 + 108162 T + p^{9} T^{2} \) |
| 19 | \( 1 + 593084 T + p^{9} T^{2} \) |
| 23 | \( 1 + 969480 T + p^{9} T^{2} \) |
| 29 | \( 1 + 6642522 T + p^{9} T^{2} \) |
| 31 | \( 1 + 7070600 T + p^{9} T^{2} \) |
| 37 | \( 1 + 7472410 T + p^{9} T^{2} \) |
| 41 | \( 1 - 4350150 T + p^{9} T^{2} \) |
| 43 | \( 1 + 4358716 T + p^{9} T^{2} \) |
| 47 | \( 1 + 28309248 T + p^{9} T^{2} \) |
| 53 | \( 1 - 16111710 T + p^{9} T^{2} \) |
| 59 | \( 1 - 86075964 T + p^{9} T^{2} \) |
| 61 | \( 1 + 32213918 T + p^{9} T^{2} \) |
| 67 | \( 1 - 99531452 T + p^{9} T^{2} \) |
| 71 | \( 1 + 44170488 T + p^{9} T^{2} \) |
| 73 | \( 1 - 23560630 T + p^{9} T^{2} \) |
| 79 | \( 1 + 401754760 T + p^{9} T^{2} \) |
| 83 | \( 1 - 744528708 T + p^{9} T^{2} \) |
| 89 | \( 1 + 769871034 T + p^{9} T^{2} \) |
| 97 | \( 1 + 907130882 T + p^{9} T^{2} \) |
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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
−10.52940843800549711902949783222, −9.561117135266530267189788307193, −8.872431147693873109760796791976, −8.074365384861084431857814009172, −6.86633143942265638915824363725, −5.81375243797427106982314310852, −3.82480187045516027731025113241, −2.12610684315710498939760074423, −1.52312240884483436462175182500, 0,
1.52312240884483436462175182500, 2.12610684315710498939760074423, 3.82480187045516027731025113241, 5.81375243797427106982314310852, 6.86633143942265638915824363725, 8.074365384861084431857814009172, 8.872431147693873109760796791976, 9.561117135266530267189788307193, 10.52940843800549711902949783222
