| L(s) = 1 | + 2·2-s − 2·3-s + 3·4-s − 3·5-s − 4·6-s − 2·7-s + 4·8-s + 3·9-s − 6·10-s + 8·11-s − 6·12-s − 3·13-s − 4·14-s + 6·15-s + 5·16-s + 2·17-s + 6·18-s + 8·19-s − 9·20-s + 4·21-s + 16·22-s − 2·23-s − 8·24-s + 5·25-s − 6·26-s − 4·27-s − 6·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 3/2·4-s − 1.34·5-s − 1.63·6-s − 0.755·7-s + 1.41·8-s + 9-s − 1.89·10-s + 2.41·11-s − 1.73·12-s − 0.832·13-s − 1.06·14-s + 1.54·15-s + 5/4·16-s + 0.485·17-s + 1.41·18-s + 1.83·19-s − 2.01·20-s + 0.872·21-s + 3.41·22-s − 0.417·23-s − 1.63·24-s + 25-s − 1.17·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.348587856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.348587856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 3 T + 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 13 T + 92 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 96 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 74 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 94 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 18 T + 190 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 14 T + 162 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 228 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.16409795382734274994877722068, −10.09124647598437864200465143322, −9.366345500486942964371074804205, −9.286258020619526811026718926491, −8.383255742330188319790166595633, −7.941888795351775698110968979097, −7.42203531196913985260854024334, −7.05251170518586249791462921764, −6.70623394113013154760483079338, −6.41104774704304517006593120815, −5.84065065123578747693183381305, −5.54959673179601220313587048257, −4.72069752741763663242035470103, −4.58582437035830247557478824988, −3.91111775644876576270518825142, −3.83468909194410644749998258392, −3.04447754182867863932522889482, −2.63902010255096877862274483980, −1.25781928333410151899553950102, −0.880648224747798634659281415318,
0.880648224747798634659281415318, 1.25781928333410151899553950102, 2.63902010255096877862274483980, 3.04447754182867863932522889482, 3.83468909194410644749998258392, 3.91111775644876576270518825142, 4.58582437035830247557478824988, 4.72069752741763663242035470103, 5.54959673179601220313587048257, 5.84065065123578747693183381305, 6.41104774704304517006593120815, 6.70623394113013154760483079338, 7.05251170518586249791462921764, 7.42203531196913985260854024334, 7.941888795351775698110968979097, 8.383255742330188319790166595633, 9.286258020619526811026718926491, 9.366345500486942964371074804205, 10.09124647598437864200465143322, 10.16409795382734274994877722068
