L(s) = 1 | + (−2.56 + 1.18i)2-s + (−4.81 + 1.96i)3-s + (5.20 − 6.07i)4-s + (−6.30 − 6.30i)5-s + (10.0 − 10.7i)6-s + 24.6·7-s + (−6.17 + 21.7i)8-s + (19.3 − 18.8i)9-s + (23.6 + 8.73i)10-s + (40.4 − 40.4i)11-s + (−13.1 + 39.4i)12-s + (−47.3 − 47.3i)13-s + (−63.3 + 29.1i)14-s + (42.6 + 17.9i)15-s + (−9.86 − 63.2i)16-s + 41.7i·17-s + ⋯ |
L(s) = 1 | + (−0.908 + 0.418i)2-s + (−0.926 + 0.377i)3-s + (0.650 − 0.759i)4-s + (−0.563 − 0.563i)5-s + (0.683 − 0.730i)6-s + 1.33·7-s + (−0.273 + 0.961i)8-s + (0.715 − 0.698i)9-s + (0.747 + 0.276i)10-s + (1.10 − 1.10i)11-s + (−0.315 + 0.948i)12-s + (−1.00 − 1.00i)13-s + (−1.21 + 0.557i)14-s + (0.734 + 0.309i)15-s + (−0.154 − 0.988i)16-s + 0.595i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.621451 - 0.204694i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.621451 - 0.204694i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.56 - 1.18i)T \) |
| 3 | \( 1 + (4.81 - 1.96i)T \) |
good | 5 | \( 1 + (6.30 + 6.30i)T + 125iT^{2} \) |
| 7 | \( 1 - 24.6T + 343T^{2} \) |
| 11 | \( 1 + (-40.4 + 40.4i)T - 1.33e3iT^{2} \) |
| 13 | \( 1 + (47.3 + 47.3i)T + 2.19e3iT^{2} \) |
| 17 | \( 1 - 41.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-10.6 + 10.6i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 + 53.4iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-105. + 105. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 - 3.14iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-42.1 + 42.1i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + 152.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (-221. - 221. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 - 381.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (294. + 294. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (445. - 445. i)T - 2.05e5iT^{2} \) |
| 61 | \( 1 + (-21.8 - 21.8i)T + 2.26e5iT^{2} \) |
| 67 | \( 1 + (572. - 572. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 612. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 331. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 427. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-245. - 245. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 188.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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
−15.33365927201137304511906418992, −14.40391979781858909860715818808, −12.17185665395609442355146023914, −11.37847562615402458695888193186, −10.35679815320995743031309200919, −8.814515679544988744281699141298, −7.74955618374135518841051891490, −6.03721782111144568750751528116, −4.68475560781677826371995358711, −0.831403935934189362623837635489,
1.71212502528242644456463510179, 4.53677181361347909741572268118, 6.92929818762965508358737187809, 7.58225740026185760789466340759, 9.387239104280609739131046959397, 10.79272637329080172591790302141, 11.75633073494532946065831501654, 12.16132348125247571944350965156, 14.26552742050003493519055859013, 15.47561721945141882904382969201