L(s) = 1 | + 16.5·5-s + (−11.9 + 14.1i)7-s − 53.4i·11-s − 12.7i·13-s − 96.7·17-s + 54.6i·19-s − 64.2i·23-s + 150.·25-s + 130. i·29-s + 220. i·31-s + (−198. + 234. i)35-s + 279.·37-s + 370.·41-s + 307.·43-s − 327.·47-s + ⋯ |
L(s) = 1 | + 1.48·5-s + (−0.644 + 0.764i)7-s − 1.46i·11-s − 0.272i·13-s − 1.38·17-s + 0.660i·19-s − 0.582i·23-s + 1.20·25-s + 0.833i·29-s + 1.27i·31-s + (−0.957 + 1.13i)35-s + 1.24·37-s + 1.41·41-s + 1.09·43-s − 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.459340263\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.459340263\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (11.9 - 14.1i)T \) |
good | 5 | \( 1 - 16.5T + 125T^{2} \) |
| 11 | \( 1 + 53.4iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 12.7iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 96.7T + 4.91e3T^{2} \) |
| 19 | \( 1 - 54.6iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 64.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 130. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 220. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 279.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 370.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 307.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 327.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 451. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 517.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 270. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 741.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 914. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 337. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 997.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 34.0T + 5.71e5T^{2} \) |
| 89 | \( 1 + 208.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.27e3iT - 9.12e5T^{2} \) |
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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
−8.852605235727104978007355757659, −8.004573495256868090710035473659, −6.62038018350126953655886044353, −6.22184158972482292900698490927, −5.64488344286343939643208929484, −4.79664337097584384908197427018, −3.41910436568934455194020868424, −2.66540911520666829874380702056, −1.82670937396879043011376896789, −0.55064687047889212799302337750,
0.881449576501817120866296330813, 2.10536717345081819774075898564, 2.57367258170363855533740806349, 4.14328187776743018406640504426, 4.61201826738446605626529148587, 5.80832403903324686399377709285, 6.40270805401107240988202437746, 7.09262427775124315744848383510, 7.81300248985850886169787264570, 9.294609820879210887106088453640