| L(s) = 1 | − 7.07i·5-s + 10.4i·7-s + (−13.0 − 34.0i)11-s + 43.7i·13-s − 115.·17-s − 89.3i·19-s + 213. i·23-s + 75·25-s − 124.·29-s + 149.·31-s + 74.0·35-s + 161.·37-s + 172.·41-s − 258. i·43-s + 240. i·47-s + ⋯ |
| L(s) = 1 | − 0.632i·5-s + 0.565i·7-s + (−0.358 − 0.933i)11-s + 0.932i·13-s − 1.64·17-s − 1.07i·19-s + 1.93i·23-s + 0.599·25-s − 0.794·29-s + 0.863·31-s + 0.357·35-s + 0.718·37-s + 0.655·41-s − 0.915i·43-s + 0.746i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.718515400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718515400\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (13.0 + 34.0i)T \) |
| good | 5 | \( 1 + 7.07iT - 125T^{2} \) |
| 7 | \( 1 - 10.4iT - 343T^{2} \) |
| 13 | \( 1 - 43.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 89.3iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 213. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 124.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 149.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 161.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 172.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 258. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 240. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 334. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 382. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 609. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 260T + 3.00e5T^{2} \) |
| 71 | \( 1 + 17.3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 787. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 1.00e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 183.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 11.1iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.79e3T + 9.12e5T^{2} \) |
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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
−9.077743843440358776491332436373, −8.450617945296208377846163080350, −7.42833818355870176968839211687, −6.56585485677975620620591576338, −5.70035566774127100299343282197, −4.90374976619862200202948958789, −4.06793423149993843509718488141, −2.86622465931618096063060486825, −1.89566026897872690270131315116, −0.60447644684343254246441405314,
0.63554898169851134851581969337, 2.13568900946397547578514621308, 2.90932818050755766942528050050, 4.15855984448352421447694970400, 4.74756168587889741598493034118, 6.00961407847923718738913192784, 6.70005258389111488071565589962, 7.46050267577995346182218642942, 8.188105509248620345716001955493, 9.079158339672723122498262116233
