L(s) = 1 | − 3.41·5-s + 4.82i·7-s − 2.82i·11-s + 2.82i·13-s + 5.41i·17-s + 5.65·19-s + 1.17·23-s + 6.65·25-s + 0.585·29-s + 3.17i·31-s − 16.4i·35-s − 3.65i·37-s − 2.58i·41-s − 9.65·43-s − 12.4·47-s + ⋯ |
L(s) = 1 | − 1.52·5-s + 1.82i·7-s − 0.852i·11-s + 0.784i·13-s + 1.31i·17-s + 1.29·19-s + 0.244·23-s + 1.33·25-s + 0.108·29-s + 0.569i·31-s − 2.78i·35-s − 0.601i·37-s − 0.403i·41-s − 1.47·43-s − 1.82·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4733675656\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4733675656\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 4.82iT - 7T^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 5.41iT - 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 - 0.585T + 29T^{2} \) |
| 31 | \( 1 - 3.17iT - 31T^{2} \) |
| 37 | \( 1 + 3.65iT - 37T^{2} \) |
| 41 | \( 1 + 2.58iT - 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 5.07T + 53T^{2} \) |
| 59 | \( 1 - 2.34iT - 59T^{2} \) |
| 61 | \( 1 - 7.65iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 6.48iT - 79T^{2} \) |
| 83 | \( 1 - 5.17iT - 83T^{2} \) |
| 89 | \( 1 + 12.2iT - 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−9.043015560438691105618740274657, −8.633592337455431045670421170740, −8.075329530220137729099340422350, −7.18407096252007300932691348954, −6.23085779363202262461065026688, −5.50713345873203090246182594790, −4.62399325594692857105404542407, −3.56164415539871279270975546360, −2.97626125278471361670244388285, −1.61906704823410697971713453222,
0.18672270352122442374392895860, 1.16800472445412204668006113520, 3.08080619841835935010827420982, 3.63113259690906198836116673394, 4.59788664525257743395550151961, 5.01980469276290794111066793055, 6.64494979224187805157744345213, 7.29748543242956014123825506728, 7.65117196171814383846528477061, 8.270739869954895807921669150189