| L(s) = 1 | − 2-s + 4·3-s − 2·4-s − 4·6-s + 6·7-s + 3·8-s + 6·9-s − 8·12-s + 7·13-s − 6·14-s + 16-s + 6·17-s − 6·18-s − 8·19-s + 24·21-s + 9·23-s + 12·24-s − 7·26-s − 4·27-s − 12·28-s − 12·29-s + 4·31-s − 2·32-s − 6·34-s − 12·36-s + 8·37-s + 8·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 2.30·3-s − 4-s − 1.63·6-s + 2.26·7-s + 1.06·8-s + 2·9-s − 2.30·12-s + 1.94·13-s − 1.60·14-s + 1/4·16-s + 1.45·17-s − 1.41·18-s − 1.83·19-s + 5.23·21-s + 1.87·23-s + 2.44·24-s − 1.37·26-s − 0.769·27-s − 2.26·28-s − 2.22·29-s + 0.718·31-s − 0.353·32-s − 1.02·34-s − 2·36-s + 1.31·37-s + 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43890625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43890625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.434217396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.434217396\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 53 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 7 T + 37 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 8 T + 49 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 9 T + 65 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_4$ | \( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 61 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 45 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 85 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18 T + 170 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 9 T + 107 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 - 19 T + 201 T^{2} - 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 7 T + 135 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 11 T + 171 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 26 T + 310 T^{2} - 26 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 14 T + 187 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + T + 177 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 123 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−8.217809977843348566202292491787, −8.202043153680197100933266133889, −7.59614306702502226142639201108, −7.58669058296825829985859250328, −7.16347688832835504739376662851, −6.51750505897872316506704076102, −5.81898833120606206140197821051, −5.76457560124844409578577722871, −5.24341384610602293943827823131, −4.94812902798819171968759941482, −4.23540117102330734942719047465, −4.10236377949016285044495581632, −3.79114180998284360835305502249, −3.58735768010091378108667049994, −2.76380983265506087445793196786, −2.49951569377338962611975792090, −2.08678634547706335308153414106, −1.61127789935772159458958463993, −0.948518392074675859667107174608, −0.866329119793624064542125792932,
0.866329119793624064542125792932, 0.948518392074675859667107174608, 1.61127789935772159458958463993, 2.08678634547706335308153414106, 2.49951569377338962611975792090, 2.76380983265506087445793196786, 3.58735768010091378108667049994, 3.79114180998284360835305502249, 4.10236377949016285044495581632, 4.23540117102330734942719047465, 4.94812902798819171968759941482, 5.24341384610602293943827823131, 5.76457560124844409578577722871, 5.81898833120606206140197821051, 6.51750505897872316506704076102, 7.16347688832835504739376662851, 7.58669058296825829985859250328, 7.59614306702502226142639201108, 8.202043153680197100933266133889, 8.217809977843348566202292491787
