| L(s) = 1 | − 3.46·5-s − 2i·7-s + 13.8i·11-s − 20.7·13-s + 18·17-s + 20.7i·19-s − 36i·23-s − 13.0·25-s + 31.1·29-s + 22i·31-s + 6.92i·35-s − 41.5·37-s − 54·41-s + 20.7i·43-s + 36i·47-s + ⋯ |
| L(s) = 1 | − 0.692·5-s − 0.285i·7-s + 1.25i·11-s − 1.59·13-s + 1.05·17-s + 1.09i·19-s − 1.56i·23-s − 0.520·25-s + 1.07·29-s + 0.709i·31-s + 0.197i·35-s − 1.12·37-s − 1.31·41-s + 0.483i·43-s + 0.765i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7653584255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7653584255\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 3.46T + 25T^{2} \) |
| 7 | \( 1 + 2iT - 49T^{2} \) |
| 11 | \( 1 - 13.8iT - 121T^{2} \) |
| 13 | \( 1 + 20.7T + 169T^{2} \) |
| 17 | \( 1 - 18T + 289T^{2} \) |
| 19 | \( 1 - 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 36iT - 529T^{2} \) |
| 29 | \( 1 - 31.1T + 841T^{2} \) |
| 31 | \( 1 - 22iT - 961T^{2} \) |
| 37 | \( 1 + 41.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 54T + 1.68e3T^{2} \) |
| 43 | \( 1 - 20.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 36iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 100.T + 2.80e3T^{2} \) |
| 59 | \( 1 + 62.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 62.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 108iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 + 50iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 13.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 18T + 7.92e3T^{2} \) |
| 97 | \( 1 + 34T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.458688287175403493277218622212, −7.78512001137190739522670378081, −7.20853209386458058254972965550, −6.50212622654195320523694827249, −5.24181586063049763988165489289, −4.64197612385890081912801446061, −3.81912763757014927954831060414, −2.76178910322190003686337738467, −1.70325571006704654315042559556, −0.23471710417514245836416505578,
0.865188031110148800678336184681, 2.39489635718795690160518039094, 3.26081119285340019929505601886, 4.07589132168543750839261375208, 5.27466707646920596281110672966, 5.60814377922112233866507038295, 6.95771638971088731836408038095, 7.40778644498076988719906201678, 8.290398317729467645431613820676, 8.878547636201544651815343686830
