| L(s) = 1 | − 16·2-s + 256·4-s − 1.17e3·5-s − 1.14e4·7-s − 4.09e3·8-s + 1.88e4·10-s + 5.84e4·11-s − 1.06e5·13-s + 1.83e5·14-s + 6.55e4·16-s − 5.93e5·17-s − 2.10e5·19-s − 3.01e5·20-s − 9.35e5·22-s + 2.08e6·23-s − 5.70e5·25-s + 1.69e6·26-s − 2.93e6·28-s + 2.39e6·29-s − 1.18e6·31-s − 1.04e6·32-s + 9.49e6·34-s + 1.34e7·35-s + 1.15e7·37-s + 3.37e6·38-s + 4.81e6·40-s + 2.39e7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.841·5-s − 1.80·7-s − 0.353·8-s + 0.595·10-s + 1.20·11-s − 1.03·13-s + 1.27·14-s + 1/4·16-s − 1.72·17-s − 0.371·19-s − 0.420·20-s − 0.851·22-s + 1.55·23-s − 0.291·25-s + 0.729·26-s − 0.903·28-s + 0.629·29-s − 0.231·31-s − 0.176·32-s + 1.21·34-s + 1.51·35-s + 1.01·37-s + 0.262·38-s + 0.297·40-s + 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.5795456963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5795456963\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 1176 T + p^{9} T^{2} \) |
| 7 | \( 1 + 1639 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 58488 T + p^{9} T^{2} \) |
| 13 | \( 1 + 8167 p T + p^{9} T^{2} \) |
| 17 | \( 1 + 593352 T + p^{9} T^{2} \) |
| 19 | \( 1 + 210967 T + p^{9} T^{2} \) |
| 23 | \( 1 - 2087832 T + p^{9} T^{2} \) |
| 29 | \( 1 - 2399424 T + p^{9} T^{2} \) |
| 31 | \( 1 + 1188772 T + p^{9} T^{2} \) |
| 37 | \( 1 - 11578187 T + p^{9} T^{2} \) |
| 41 | \( 1 - 23941632 T + p^{9} T^{2} \) |
| 43 | \( 1 + 10659832 T + p^{9} T^{2} \) |
| 47 | \( 1 - 34054008 T + p^{9} T^{2} \) |
| 53 | \( 1 + 42741072 T + p^{9} T^{2} \) |
| 59 | \( 1 - 74207928 T + p^{9} T^{2} \) |
| 61 | \( 1 - 81024095 T + p^{9} T^{2} \) |
| 67 | \( 1 + 19650859 T + p^{9} T^{2} \) |
| 71 | \( 1 + 2594160 p T + p^{9} T^{2} \) |
| 73 | \( 1 + 257037703 T + p^{9} T^{2} \) |
| 79 | \( 1 - 651592289 T + p^{9} T^{2} \) |
| 83 | \( 1 + 17638608 T + p^{9} T^{2} \) |
| 89 | \( 1 + 516254760 T + p^{9} T^{2} \) |
| 97 | \( 1 + 434232691 T + p^{9} T^{2} \) |
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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
−13.15075685067526550866926591018, −12.17534778465273162341147412363, −11.04063896152594944979070476889, −9.627452199498378734611432044184, −8.893920076699268629284666386152, −7.18067878122469158788479330554, −6.43193391783913133004855321405, −4.12333724552426205023978324628, −2.70116853980517758509504741349, −0.51443664285720833742892271271,
0.51443664285720833742892271271, 2.70116853980517758509504741349, 4.12333724552426205023978324628, 6.43193391783913133004855321405, 7.18067878122469158788479330554, 8.893920076699268629284666386152, 9.627452199498378734611432044184, 11.04063896152594944979070476889, 12.17534778465273162341147412363, 13.15075685067526550866926591018
