| L(s) = 1 | + (−2.90 + 5.02i)5-s + (13.0 + 22.6i)7-s + (−18.6 − 32.2i)11-s + (15.1 − 26.2i)13-s + 48.4·17-s + 88.1·19-s + (42.2 − 73.2i)23-s + (45.6 + 79.0i)25-s + (88.3 + 153. i)29-s + (−77.6 + 134. i)31-s − 151.·35-s − 258.·37-s + (110. − 190. i)41-s + (23.8 + 41.3i)43-s + (64.7 + 112. i)47-s + ⋯ |
| L(s) = 1 | + (−0.259 + 0.449i)5-s + (0.705 + 1.22i)7-s + (−0.510 − 0.884i)11-s + (0.323 − 0.560i)13-s + 0.691·17-s + 1.06·19-s + (0.383 − 0.663i)23-s + (0.365 + 0.632i)25-s + (0.565 + 0.980i)29-s + (−0.450 + 0.779i)31-s − 0.732·35-s − 1.14·37-s + (0.419 − 0.726i)41-s + (0.0846 + 0.146i)43-s + (0.200 + 0.348i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.995636552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.995636552\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (2.90 - 5.02i)T + (-62.5 - 108. i)T^{2} \) |
| 7 | \( 1 + (-13.0 - 22.6i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (18.6 + 32.2i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-15.1 + 26.2i)T + (-1.09e3 - 1.90e3i)T^{2} \) |
| 17 | \( 1 - 48.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 88.1T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-42.2 + 73.2i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-88.3 - 153. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (77.6 - 134. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-110. + 190. i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-23.8 - 41.3i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-64.7 - 112. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + 577.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (121. - 210. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-343. - 595. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (189. - 327. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 332.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + (-630. - 1.09e3i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-613. - 1.06e3i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-564. - 977. i)T + (-4.56e5 + 7.90e5i)T^{2} \) |
| 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
−10.62878210267213101126655197733, −9.294504512396722636309940221111, −8.535290877081150618517995744820, −7.84690644052181243892444365040, −6.80005449160615779797446257364, −5.52021496537621699116592281510, −5.19160352560398816517716221292, −3.43753763334217692787362969335, −2.71379222498787662810446662400, −1.15930015740232643284848692561,
0.67034602112233554670572247768, 1.79477573936475176740690458952, 3.46090545493900756364223244751, 4.49452165585186382183337227992, 5.14033917146779235002067205582, 6.53771261922309914811596223802, 7.64716474927651356562981089511, 7.88541249589828052408416192366, 9.224256194983513941322867942584, 10.00396725372395249880514456539
