| L(s) = 1 | − 7.06·5-s + (−18.4 + 1.86i)7-s + 8.54i·11-s + 52.3i·13-s + 38.9·17-s − 66.4i·19-s − 200. i·23-s − 75.0·25-s + 61.1i·29-s − 134. i·31-s + (130. − 13.2i)35-s + 298.·37-s + 442.·41-s − 52.2·43-s − 275.·47-s + ⋯ |
| L(s) = 1 | − 0.632·5-s + (−0.994 + 0.100i)7-s + 0.234i·11-s + 1.11i·13-s + 0.555·17-s − 0.801i·19-s − 1.82i·23-s − 0.600·25-s + 0.391i·29-s − 0.778i·31-s + (0.629 − 0.0637i)35-s + 1.32·37-s + 1.68·41-s − 0.185·43-s − 0.854·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8186011694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8186011694\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (18.4 - 1.86i)T \) |
| good | 5 | \( 1 + 7.06T + 125T^{2} \) |
| 11 | \( 1 - 8.54iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 52.3iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 38.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 66.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 200. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 61.1iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 134. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 298.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 442.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 275.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 136. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 382.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 302. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 637.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 228. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 1.24e3iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 201.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 646.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 826.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 19.7iT - 9.12e5T^{2} \) |
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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
−9.085691437139291782341507709575, −8.020926680256074539402162812355, −7.34129731627604652616457237345, −6.50508575659057021357683158891, −5.97816607453520320654488960593, −4.62834053739192213749841359276, −4.13988762987308812594392539019, −3.07470323249743577290389718297, −2.23732280642453028395619685541, −0.74984882705725707986883728022,
0.23464882905022463748777446873, 1.32985669550207890888632628184, 2.86478400910010016841878038430, 3.47227242273483064652235368007, 4.22830010782837043007512409170, 5.57754602755722813480949157874, 5.91764655367095472575665585746, 7.04475618282747095675097949833, 7.79537836776880174111730419600, 8.213228861551801833529180367098
