| L(s) = 1 | + (4.26 − 2.61i)5-s − 5.45i·7-s + 7.38i·11-s + 15.9i·13-s + 15.3·17-s − 2.76·19-s + 3.49·23-s + (11.3 − 22.2i)25-s + 43.6i·29-s − 1.13·31-s + (−14.2 − 23.2i)35-s + 47.5i·37-s + 6.64i·41-s + 23.1i·43-s − 39.6·47-s + ⋯ |
| L(s) = 1 | + (0.852 − 0.522i)5-s − 0.779i·7-s + 0.670i·11-s + 1.23i·13-s + 0.900·17-s − 0.145·19-s + 0.151·23-s + (0.453 − 0.891i)25-s + 1.50i·29-s − 0.0365·31-s + (−0.407 − 0.664i)35-s + 1.28i·37-s + 0.162i·41-s + 0.539i·43-s − 0.844·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.394092375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.394092375\) |
| \(L(2)\) |
|
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 | \( 1 + (-4.26 + 2.61i)T \) |
| good | 7 | \( 1 + 5.45iT - 49T^{2} \) |
| 11 | \( 1 - 7.38iT - 121T^{2} \) |
| 13 | \( 1 - 15.9iT - 169T^{2} \) |
| 17 | \( 1 - 15.3T + 289T^{2} \) |
| 19 | \( 1 + 2.76T + 361T^{2} \) |
| 23 | \( 1 - 3.49T + 529T^{2} \) |
| 29 | \( 1 - 43.6iT - 841T^{2} \) |
| 31 | \( 1 + 1.13T + 961T^{2} \) |
| 37 | \( 1 - 47.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 6.64iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 23.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 39.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 52.1T + 2.80e3T^{2} \) |
| 59 | \( 1 - 49.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 39.7T + 3.72e3T^{2} \) |
| 67 | \( 1 - 71.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 85.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 28.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 90.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 97.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 121. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 77.9iT - 9.40e3T^{2} \) |
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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
−9.036544475211408717170809699318, −8.295614401106144430124580457806, −7.23284093778763854323291753290, −6.73969529694860529884349136775, −5.78653984033138311192629281259, −4.86518756901221003654950662431, −4.27230391236528859812479265031, −3.10882312592180583334986222989, −1.84605986992610680855436365983, −1.11160069731174358898775374359,
0.64729526827256238641799407595, 2.06060749311273513694059807903, 2.86888498102608667028837144881, 3.67824895067183570530447981452, 5.13409825174955149010391603215, 5.75722889788412244984041538396, 6.19867094093026955895852058458, 7.32836745476551530406538733576, 8.078195247171008443799194582110, 8.859541455912937921542051911372
