L(s) = 1 | + (−2.18 − 1.79i)2-s + (−2.12 − 2.12i)3-s + (1.55 + 7.84i)4-s + (−3.87 − 10.4i)5-s + (0.829 + 8.44i)6-s + (−17.0 + 17.0i)7-s + (10.6 − 19.9i)8-s + 8.99i·9-s + (−10.3 + 29.8i)10-s + 62.6i·11-s + (13.3 − 19.9i)12-s + (−10.3 + 10.3i)13-s + (67.8 − 6.66i)14-s + (−14.0 + 30.4i)15-s + (−59.1 + 24.4i)16-s + (−15.8 − 15.8i)17-s + ⋯ |
L(s) = 1 | + (−0.772 − 0.634i)2-s + (−0.408 − 0.408i)3-s + (0.194 + 0.980i)4-s + (−0.346 − 0.937i)5-s + (0.0564 + 0.574i)6-s + (−0.919 + 0.919i)7-s + (0.472 − 0.881i)8-s + 0.333i·9-s + (−0.327 + 0.945i)10-s + 1.71i·11-s + (0.320 − 0.479i)12-s + (−0.221 + 0.221i)13-s + (1.29 − 0.127i)14-s + (−0.241 + 0.524i)15-s + (−0.924 + 0.381i)16-s + (−0.225 − 0.225i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.312 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.0781722 + 0.108068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0781722 + 0.108068i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (2.18 + 1.79i)T \) |
| 3 | \( 1 + (2.12 + 2.12i)T \) |
| 5 | \( 1 + (3.87 + 10.4i)T \) |
good | 7 | \( 1 + (17.0 - 17.0i)T - 343iT^{2} \) |
| 11 | \( 1 - 62.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (10.3 - 10.3i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (15.8 + 15.8i)T + 4.91e3iT^{2} \) |
| 19 | \( 1 + 40.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + (144. + 144. i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 97.7iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 3.67iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (-158. - 158. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 + (59.6 + 59.6i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (0.929 - 0.929i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (385. - 385. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 - 112.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 420.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (505. - 505. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 + 713. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-541. + 541. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 277.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (96.5 + 96.5i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 370. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-701. - 701. i)T + 9.12e5iT^{2} \) |
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
−15.29906824472997011112463701394, −13.14955593174053735114140610663, −12.31484506904208294008348940954, −11.92797513813453057154785088416, −10.12591416039952494041028276330, −9.207857636548860289771275240530, −7.976433600447468390708105225274, −6.54832228358927988588871082289, −4.49667625039761446357673701751, −2.15174549456702426646957192414,
0.11922224329422043465973957863, 3.58181365969681803664403008943, 5.87621179926017465107922924522, 6.83680239220953092722840258179, 8.159746127576179025028795760045, 9.742505613209024727145813260330, 10.61986347371309950776440993740, 11.42687710060179105445782843142, 13.49221135760942702834515182445, 14.46476320096090563536992404030