| L(s) = 1 | − 512·2-s − 1.96e4·3-s + 2.62e5·4-s − 3.73e6·5-s + 1.00e7·6-s − 1.49e8·7-s − 1.34e8·8-s + 3.87e8·9-s + 1.91e9·10-s − 7.45e9·11-s − 5.15e9·12-s + 5.92e10·13-s + 7.66e10·14-s + 7.34e10·15-s + 6.87e10·16-s + 5.23e11·17-s − 1.98e11·18-s + 9.69e11·19-s − 9.78e11·20-s + 2.94e12·21-s + 3.81e12·22-s − 1.36e12·23-s + 2.64e12·24-s − 5.14e12·25-s − 3.03e13·26-s − 7.62e12·27-s − 3.92e13·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.854·5-s + 0.408·6-s − 1.40·7-s − 0.353·8-s + 1/3·9-s + 0.604·10-s − 0.953·11-s − 0.288·12-s + 1.54·13-s + 0.991·14-s + 0.493·15-s + 1/4·16-s + 1.06·17-s − 0.235·18-s + 0.689·19-s − 0.427·20-s + 0.809·21-s + 0.674·22-s − 0.158·23-s + 0.204·24-s − 0.269·25-s − 1.09·26-s − 0.192·27-s − 0.700·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(0.6127783298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6127783298\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{9} T \) |
| 3 | \( 1 + p^{9} T \) |
| good | 5 | \( 1 + 3732474 T + p^{19} T^{2} \) |
| 7 | \( 1 + 3054544 p^{2} T + p^{19} T^{2} \) |
| 11 | \( 1 + 678152028 p T + p^{19} T^{2} \) |
| 13 | \( 1 - 4556804606 p T + p^{19} T^{2} \) |
| 17 | \( 1 - 30771201762 p T + p^{19} T^{2} \) |
| 19 | \( 1 - 969502037780 T + p^{19} T^{2} \) |
| 23 | \( 1 + 1368374071512 T + p^{19} T^{2} \) |
| 29 | \( 1 + 98642915804130 T + p^{19} T^{2} \) |
| 31 | \( 1 - 194951985476072 T + p^{19} T^{2} \) |
| 37 | \( 1 - 1187317903389374 T + p^{19} T^{2} \) |
| 41 | \( 1 - 1870198963153962 T + p^{19} T^{2} \) |
| 43 | \( 1 + 148368687075892 T + p^{19} T^{2} \) |
| 47 | \( 1 + 10572302364578016 T + p^{19} T^{2} \) |
| 53 | \( 1 + 36343891942173882 T + p^{19} T^{2} \) |
| 59 | \( 1 - 41470141982543340 T + p^{19} T^{2} \) |
| 61 | \( 1 - 147497582335087142 T + p^{19} T^{2} \) |
| 67 | \( 1 + 373863557053725196 T + p^{19} T^{2} \) |
| 71 | \( 1 - 682838097709300632 T + p^{19} T^{2} \) |
| 73 | \( 1 + 69476440231543462 T + p^{19} T^{2} \) |
| 79 | \( 1 - 408504634469931320 T + p^{19} T^{2} \) |
| 83 | \( 1 + 199775389551817452 T + p^{19} T^{2} \) |
| 89 | \( 1 - 1270681646000976810 T + p^{19} T^{2} \) |
| 97 | \( 1 - 2514259692302428034 T + p^{19} T^{2} \) |
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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
−18.30549464967954718814637928761, −16.34922735681804201144662924746, −15.73674211013459842682455114090, −12.97029014384744057352251090480, −11.37827827169201235526745402445, −9.842298639605909808383832267722, −7.82962382761782684132632563604, −6.07923107406356974171131941099, −3.39678010518349663072931978746, −0.65955730916605642386872548095,
0.65955730916605642386872548095, 3.39678010518349663072931978746, 6.07923107406356974171131941099, 7.82962382761782684132632563604, 9.842298639605909808383832267722, 11.37827827169201235526745402445, 12.97029014384744057352251090480, 15.73674211013459842682455114090, 16.34922735681804201144662924746, 18.30549464967954718814637928761
