L(s) = 1 | − i·5-s + 4.73·7-s − 3.46i·11-s − 3.46i·13-s + 3.46·17-s − 2i·19-s + 2.19·23-s − 25-s − 2.53·31-s − 4.73i·35-s − 6i·37-s − 9.46·41-s − 0.196i·43-s − 2.19·47-s + 15.3·49-s + ⋯ |
L(s) = 1 | − 0.447i·5-s + 1.78·7-s − 1.04i·11-s − 0.960i·13-s + 0.840·17-s − 0.458i·19-s + 0.457·23-s − 0.200·25-s − 0.455·31-s − 0.799i·35-s − 0.986i·37-s − 1.47·41-s − 0.0299i·43-s − 0.320·47-s + 2.19·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.260701655\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.260701655\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
good | 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 + 0.196iT - 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 0.928iT - 61T^{2} \) |
| 67 | \( 1 + 0.196iT - 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 1.26iT - 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−8.623627749572606350101518978206, −7.84227679801819283806266207912, −7.46478658287170150840018942439, −6.14029630052336634648607145751, −5.31710967860216297578110082162, −4.97020720747110824894902887856, −3.88200187039545453162701796307, −2.91253215631202599062806034442, −1.67537820467108124592958741136, −0.76104772233906374179756945826,
1.48743359418671449145646546010, 2.02130523769817301593823351542, 3.33549009450246415815202602284, 4.41992202938802483792358204239, 4.90440615162653628871812081313, 5.77509651081506931069624856806, 6.89669638307692981804733847675, 7.38353231088624702540139069927, 8.163828778310000269990804204666, 8.767142687878435866037367964349