| L(s) = 1 | + 3·2-s + 3·3-s + 4·4-s − 3·5-s + 9·6-s + 3·8-s + 2·9-s − 9·10-s − 3·11-s + 12·12-s − 9·15-s + 3·16-s − 6·17-s + 6·18-s − 3·19-s − 12·20-s − 9·22-s + 9·24-s − 2·25-s − 6·27-s + 3·29-s − 27·30-s − 4·31-s + 6·32-s − 9·33-s − 18·34-s + 8·36-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 2·4-s − 1.34·5-s + 3.67·6-s + 1.06·8-s + 2/3·9-s − 2.84·10-s − 0.904·11-s + 3.46·12-s − 2.32·15-s + 3/4·16-s − 1.45·17-s + 1.41·18-s − 0.688·19-s − 2.68·20-s − 1.91·22-s + 1.83·24-s − 2/5·25-s − 1.15·27-s + 0.557·29-s − 4.92·30-s − 0.718·31-s + 1.06·32-s − 1.56·33-s − 3.08·34-s + 4/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68574961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.757067098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.757067098\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T + 5 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - p T + 7 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 3 T + 11 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3 T + 13 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + p T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 86 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 5 T - 9 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 89 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 137 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 113 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 125 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 121 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 257 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 31 T + 423 T^{2} + 31 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.084102582488861026432787061015, −7.62127837189427387435719531455, −7.21155323685210983855805943042, −7.14243089393315391885520254720, −6.34707201148660813755976352196, −6.34281375708795737053787227609, −5.64843928970997031583735685192, −5.44870644659982544770940746533, −5.14236149741072595125046710710, −4.52660079780271122509648938491, −4.23727145960997947553527993506, −4.16645100936657774096940211964, −3.63344338937490639316742861317, −3.60916752362497800212586570879, −2.90803446053398436745480412613, −2.69105054450256737213774109615, −2.42058964687171941975378548148, −2.01702888534247557345047972132, −1.24746181488089602122483371649, −0.17374342297640090644639670689,
0.17374342297640090644639670689, 1.24746181488089602122483371649, 2.01702888534247557345047972132, 2.42058964687171941975378548148, 2.69105054450256737213774109615, 2.90803446053398436745480412613, 3.60916752362497800212586570879, 3.63344338937490639316742861317, 4.16645100936657774096940211964, 4.23727145960997947553527993506, 4.52660079780271122509648938491, 5.14236149741072595125046710710, 5.44870644659982544770940746533, 5.64843928970997031583735685192, 6.34281375708795737053787227609, 6.34707201148660813755976352196, 7.14243089393315391885520254720, 7.21155323685210983855805943042, 7.62127837189427387435719531455, 8.084102582488861026432787061015
