L(s) = 1 | − 1.30i·3-s − 4.30i·7-s + 1.30·9-s + 11-s − 5i·13-s − 3.90i·17-s − 19-s − 5.60·21-s − 3.69i·23-s − 5.60i·27-s + 9.90·29-s + 4.21·31-s − 1.30i·33-s + 9.60i·37-s − 6.51·39-s + ⋯ |
L(s) = 1 | − 0.752i·3-s − 1.62i·7-s + 0.434·9-s + 0.301·11-s − 1.38i·13-s − 0.947i·17-s − 0.229·19-s − 1.22·21-s − 0.770i·23-s − 1.07i·27-s + 1.83·29-s + 0.756·31-s − 0.226i·33-s + 1.57i·37-s − 1.04·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.071073901\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.071073901\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + 1.30iT - 3T^{2} \) |
| 7 | \( 1 + 4.30iT - 7T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 + 3.90iT - 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + 3.69iT - 23T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 37 | \( 1 - 9.60iT - 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 + 7.21iT - 43T^{2} \) |
| 47 | \( 1 - 3iT - 47T^{2} \) |
| 53 | \( 1 + 2.30iT - 53T^{2} \) |
| 59 | \( 1 - 0.211T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 4.60T + 71T^{2} \) |
| 73 | \( 1 + 2.90iT - 73T^{2} \) |
| 79 | \( 1 + 0.0916T + 79T^{2} \) |
| 83 | \( 1 - 14.5iT - 83T^{2} \) |
| 89 | \( 1 + 5.30T + 89T^{2} \) |
| 97 | \( 1 - 11.6iT - 97T^{2} \) |
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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
−8.039561510314826712283770928498, −7.17468786208329908313001531958, −6.83269815134482350696774564533, −6.11044413553715349916126183037, −4.87475472329821282395741608023, −4.39015237267620016071973132426, −3.37308117511859779474210871641, −2.52173082535976043270798528711, −1.10382496678185013011507823931, −0.68386610627091710063430171487,
1.52648357535240705285244676822, 2.34768618212127408704412799144, 3.37497581786662752671429438514, 4.29948723498312921359661198126, 4.79026280665010030185415953282, 5.79541986940834833416421878507, 6.31913508663060298860209540310, 7.13226757602496250769499103254, 8.184706038497842868937869257146, 8.854349467420096433923769629788