L(s) = 1 | − 5-s − 7-s + 6·13-s + 3·17-s + 4·19-s − 6·23-s − 4·25-s − 6·31-s + 35-s − 6·37-s + 10·41-s − 11·43-s + 9·47-s + 49-s + 12·53-s − 7·59-s − 2·61-s − 6·65-s − 9·67-s + 2·71-s + 2·73-s − 4·79-s + 3·83-s − 3·85-s − 15·89-s − 6·91-s − 4·95-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.66·13-s + 0.727·17-s + 0.917·19-s − 1.25·23-s − 4/5·25-s − 1.07·31-s + 0.169·35-s − 0.986·37-s + 1.56·41-s − 1.67·43-s + 1.31·47-s + 1/7·49-s + 1.64·53-s − 0.911·59-s − 0.256·61-s − 0.744·65-s − 1.09·67-s + 0.237·71-s + 0.234·73-s − 0.450·79-s + 0.329·83-s − 0.325·85-s − 1.58·89-s − 0.628·91-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.862984509\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.862984509\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.12101341483508, −13.73041525048049, −13.49133100723893, −12.69890644077895, −12.23102070170504, −11.80841168283696, −11.27711294793416, −10.77262196624219, −10.20694889357272, −9.728691229601231, −9.128454378895268, −8.558590092055898, −8.152553021372924, −7.402857767488705, −7.204811924212936, −6.246858855257943, −5.811302925629612, −5.507151352928439, −4.534954651443044, −3.804923151841062, −3.623105215636101, −2.937325476552802, −1.968318550580053, −1.329465345252814, −0.4873494132897363,
0.4873494132897363, 1.329465345252814, 1.968318550580053, 2.937325476552802, 3.623105215636101, 3.804923151841062, 4.534954651443044, 5.507151352928439, 5.811302925629612, 6.246858855257943, 7.204811924212936, 7.402857767488705, 8.152553021372924, 8.558590092055898, 9.128454378895268, 9.728691229601231, 10.20694889357272, 10.77262196624219, 11.27711294793416, 11.80841168283696, 12.23102070170504, 12.69890644077895, 13.49133100723893, 13.73041525048049, 14.12101341483508