| L(s) = 1 | − 427. i·5-s + 679·7-s + 1.33e4i·11-s − 3.08e4·13-s + 1.28e5i·17-s + 1.38e5·19-s + 3.03e5i·23-s + 2.08e5·25-s − 1.32e6i·29-s − 3.52e5·31-s − 2.90e5i·35-s + 1.18e6·37-s + 1.09e6i·41-s − 6.24e6·43-s − 2.39e3i·47-s + ⋯ |
| L(s) = 1 | − 0.683i·5-s + 0.282·7-s + 0.913i·11-s − 1.07·13-s + 1.53i·17-s + 1.06·19-s + 1.08i·23-s + 0.532·25-s − 1.87i·29-s − 0.381·31-s − 0.193i·35-s + 0.634·37-s + 0.387i·41-s − 1.82·43-s − 0.000491i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.05451479303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05451479303\) |
| \(L(5)\) |
|
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 | \( 1 + 427. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 679T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.33e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 3.08e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 1.28e5iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.38e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 3.03e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.32e6iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 3.52e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 1.18e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.09e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 6.24e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 2.39e3iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.25e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.05e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.65e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 7.66e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.32e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.49e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 4.16e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 4.47e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 7.40e5iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.05e8T + 7.83e15T^{2} \) |
| 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
−9.906437173694253392076743852451, −9.638744878936402825602897845786, −8.313702189789435243486426011580, −7.69376881584451534001410650608, −6.61855000653405970708524871912, −5.38473009736849522244318493485, −4.69597434137136910405320367507, −3.62138968783004894433871479990, −2.18014048234423823452926011721, −1.31609899862194380377544304859,
0.01044917746919425543602614560, 1.13709470369392499243524811964, 2.65040714055657875978688035661, 3.20706263457188276983208679931, 4.73170150515198824434267481226, 5.47402530099105774438392314830, 6.81987587061456221391240696387, 7.32563281554017894365127741935, 8.482974747805600448192540019342, 9.396750868156677351548368468605
