L(s) = 1 | + 2.35·5-s + i·7-s − 1.09i·11-s + 4.10i·13-s + 4.74i·17-s + 7.10·19-s + 3.99·23-s + 0.523·25-s + 4.09·29-s − 10.8i·31-s + 2.35i·35-s − 2.27i·37-s + 1.29i·41-s + 8.59·43-s − 6.79·47-s + ⋯ |
L(s) = 1 | + 1.05·5-s + 0.377i·7-s − 0.328i·11-s + 1.13i·13-s + 1.15i·17-s + 1.62·19-s + 0.832·23-s + 0.104·25-s + 0.761·29-s − 1.95i·31-s + 0.397i·35-s − 0.373i·37-s + 0.201i·41-s + 1.31·43-s − 0.991·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.681503959\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.681503959\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 5 | \( 1 - 2.35T + 5T^{2} \) |
| 11 | \( 1 + 1.09iT - 11T^{2} \) |
| 13 | \( 1 - 4.10iT - 13T^{2} \) |
| 17 | \( 1 - 4.74iT - 17T^{2} \) |
| 19 | \( 1 - 7.10T + 19T^{2} \) |
| 23 | \( 1 - 3.99T + 23T^{2} \) |
| 29 | \( 1 - 4.09T + 29T^{2} \) |
| 31 | \( 1 + 10.8iT - 31T^{2} \) |
| 37 | \( 1 + 2.27iT - 37T^{2} \) |
| 41 | \( 1 - 1.29iT - 41T^{2} \) |
| 43 | \( 1 - 8.59T + 43T^{2} \) |
| 47 | \( 1 + 6.79T + 47T^{2} \) |
| 53 | \( 1 + 0.688T + 53T^{2} \) |
| 59 | \( 1 + 7.72iT - 59T^{2} \) |
| 61 | \( 1 - 9.70iT - 61T^{2} \) |
| 67 | \( 1 + 8.23T + 67T^{2} \) |
| 71 | \( 1 + 6.46T + 71T^{2} \) |
| 73 | \( 1 - 8.44T + 73T^{2} \) |
| 79 | \( 1 - 5.70iT - 79T^{2} \) |
| 83 | \( 1 - 10.0iT - 83T^{2} \) |
| 89 | \( 1 + 2.54iT - 89T^{2} \) |
| 97 | \( 1 + 15.6T + 97T^{2} \) |
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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
−8.180590903899528918395857549704, −7.44692779386703947765911859293, −6.58635692677520993438860938775, −5.97471748036420753789031489989, −5.49127551774026968529556371753, −4.58040504535866275112770416025, −3.73379889381815290459385974289, −2.73866407430504875371508400203, −1.97263321155274351528443343788, −1.08324308001420716943483310063,
0.77756276212792733425982193073, 1.61222537482193647574363670723, 2.93131575714013403095529084168, 3.14921503905816629506628907753, 4.59457805335263704252716374372, 5.19732945371735093676356054116, 5.70202921011116522512068100840, 6.65762047489725351325122794430, 7.25504316961551315649006348333, 7.86120616478631759643257401800