| L(s) = 1 | + 512·2-s + 5.30e4·3-s + 2.62e5·4-s + 5.55e6·5-s + 2.71e7·6-s + 1.34e8·8-s + 1.64e9·9-s + 2.84e9·10-s + 6.32e9·11-s + 1.39e10·12-s + 3.31e10·13-s + 2.94e11·15-s + 6.87e10·16-s + 7.22e11·17-s + 8.44e11·18-s + 1.31e12·19-s + 1.45e12·20-s + 3.23e12·22-s + 3.37e12·23-s + 7.11e12·24-s + 1.18e13·25-s + 1.69e13·26-s + 2.58e13·27-s − 2.93e13·29-s + 1.50e14·30-s − 1.31e14·31-s + 3.51e13·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.55·3-s + 1/2·4-s + 1.27·5-s + 1.09·6-s + 0.353·8-s + 1.41·9-s + 0.899·10-s + 0.808·11-s + 0.777·12-s + 0.866·13-s + 1.97·15-s + 1/4·16-s + 1.47·17-s + 1.00·18-s + 0.933·19-s + 0.636·20-s + 0.571·22-s + 0.391·23-s + 0.549·24-s + 0.618·25-s + 0.612·26-s + 0.652·27-s − 0.376·29-s + 1.39·30-s − 0.896·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(11.25909901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.25909901\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{9} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1964 p^{3} T + p^{19} T^{2} \) |
| 5 | \( 1 - 1111386 p T + p^{19} T^{2} \) |
| 11 | \( 1 - 574606812 p T + p^{19} T^{2} \) |
| 13 | \( 1 - 33124973098 T + p^{19} T^{2} \) |
| 17 | \( 1 - 42491485422 p T + p^{19} T^{2} \) |
| 19 | \( 1 - 1312620671860 T + p^{19} T^{2} \) |
| 23 | \( 1 - 3379752742152 T + p^{19} T^{2} \) |
| 29 | \( 1 + 29378097714810 T + p^{19} T^{2} \) |
| 31 | \( 1 + 131976476089952 T + p^{19} T^{2} \) |
| 37 | \( 1 + 466464103652194 T + p^{19} T^{2} \) |
| 41 | \( 1 + 1889447681239482 T + p^{19} T^{2} \) |
| 43 | \( 1 + 4323507451065388 T + p^{19} T^{2} \) |
| 47 | \( 1 + 12103384387771536 T + p^{19} T^{2} \) |
| 53 | \( 1 + 30593935900444338 T + p^{19} T^{2} \) |
| 59 | \( 1 + 9908742512283780 T + p^{19} T^{2} \) |
| 61 | \( 1 - 91638145794467098 T + p^{19} T^{2} \) |
| 67 | \( 1 + 103349440678278244 T + p^{19} T^{2} \) |
| 71 | \( 1 - 285448322456957592 T + p^{19} T^{2} \) |
| 73 | \( 1 + 875008267167254042 T + p^{19} T^{2} \) |
| 79 | \( 1 + 1081394522969090320 T + p^{19} T^{2} \) |
| 83 | \( 1 - 665085275193888948 T + p^{19} T^{2} \) |
| 89 | \( 1 - 2020985164277790390 T + p^{19} T^{2} \) |
| 97 | \( 1 - 12825578365118067934 T + p^{19} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.10347354368949151168933096186, −9.430962105129407568390614865438, −8.450756525992699945647466506162, −7.29862631862031661149436259100, −6.14976049926677508881237386571, −5.12741574651465806446838669676, −3.55805965854095001197582218328, −3.16376489166974324761080347650, −1.80668578182145275476438604941, −1.37918229887256030536824093465,
1.37918229887256030536824093465, 1.80668578182145275476438604941, 3.16376489166974324761080347650, 3.55805965854095001197582218328, 5.12741574651465806446838669676, 6.14976049926677508881237386571, 7.29862631862031661149436259100, 8.450756525992699945647466506162, 9.430962105129407568390614865438, 10.10347354368949151168933096186
