L(s) = 1 | − 5-s + 0.449i·7-s − 0.449i·11-s + 1.41i·13-s − 2.82i·17-s − 2.82·19-s + 3.46·23-s + 25-s + 2·29-s + 4.89i·31-s − 0.449i·35-s + 5.51i·37-s − 5.51i·41-s + 4.09·47-s + 6.79·49-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.169i·7-s − 0.135i·11-s + 0.392i·13-s − 0.685i·17-s − 0.648·19-s + 0.722·23-s + 0.200·25-s + 0.371·29-s + 0.879i·31-s − 0.0759i·35-s + 0.906i·37-s − 0.861i·41-s + 0.598·47-s + 0.971·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.769 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.478309039\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.478309039\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
good | 7 | \( 1 - 0.449iT - 7T^{2} \) |
| 11 | \( 1 + 0.449iT - 11T^{2} \) |
| 13 | \( 1 - 1.41iT - 13T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 5.51iT - 37T^{2} \) |
| 41 | \( 1 + 5.51iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 5.34iT - 59T^{2} \) |
| 61 | \( 1 - 1.55iT - 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 7.79T + 73T^{2} \) |
| 79 | \( 1 - 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 8.89iT - 83T^{2} \) |
| 89 | \( 1 - 0.142iT - 89T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
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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
−8.803791003800586269640150194961, −8.219524896497716603080465711577, −7.21325868036464351153783872028, −6.77222061921245555156077985252, −5.75824549731015290507040919019, −4.92519555314467840084562780271, −4.16266432492183794344004256868, −3.20167499063318925265639684952, −2.29146832634419190759476662080, −0.926627928347291665972406141261,
0.60919932718056149615885487039, 1.97895107213756471505625435332, 3.06085260184395142438531388991, 3.98004203256075640003927447226, 4.66892855419893912115539095638, 5.67299843092827777965897311856, 6.41131621062782310794290858404, 7.28612715521782096593319604137, 7.905154321499455300094630229204, 8.657215341722779357882310351281