| L(s) = 1 | + 2·2-s + 2·4-s + 6·7-s + 12·14-s − 4·16-s + 2·23-s + 12·28-s − 8·32-s − 24·41-s + 18·43-s + 4·46-s + 14·47-s + 18·49-s − 16·61-s − 8·64-s + 6·67-s − 48·82-s + 22·83-s + 36·86-s + 4·92-s + 28·94-s + 36·98-s + 36·101-s + 18·103-s − 26·107-s − 24·112-s + 22·121-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 2.26·7-s + 3.20·14-s − 16-s + 0.417·23-s + 2.26·28-s − 1.41·32-s − 3.74·41-s + 2.74·43-s + 0.589·46-s + 2.04·47-s + 18/7·49-s − 2.04·61-s − 64-s + 0.733·67-s − 5.30·82-s + 2.41·83-s + 3.88·86-s + 0.417·92-s + 2.88·94-s + 3.63·98-s + 3.58·101-s + 1.77·103-s − 2.51·107-s − 2.26·112-s + 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.759566818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.759566818\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
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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
−10.59099966305573573881295562982, −10.16343299952444866717723104132, −9.195762203984399164701767355218, −9.109383217348625344383205361102, −8.642548864725372871977947213908, −8.078992412830036582370007384411, −7.80471716672948027902564736696, −7.24056898474944006872413298440, −6.95046358721728681067822128429, −6.21313502162534461243512833285, −5.86220419260858020584297175769, −5.21858616755767649957089447804, −5.10961798185405734893363398190, −4.41505097194284748975876512516, −4.40357451549800893699039208209, −3.54007083781514542106327202164, −3.12451912343664057647706000647, −2.14267014497467420032854894666, −1.98528519039804122427485905946, −0.966108576971304450391810362719,
0.966108576971304450391810362719, 1.98528519039804122427485905946, 2.14267014497467420032854894666, 3.12451912343664057647706000647, 3.54007083781514542106327202164, 4.40357451549800893699039208209, 4.41505097194284748975876512516, 5.10961798185405734893363398190, 5.21858616755767649957089447804, 5.86220419260858020584297175769, 6.21313502162534461243512833285, 6.95046358721728681067822128429, 7.24056898474944006872413298440, 7.80471716672948027902564736696, 8.078992412830036582370007384411, 8.642548864725372871977947213908, 9.109383217348625344383205361102, 9.195762203984399164701767355218, 10.16343299952444866717723104132, 10.59099966305573573881295562982
