L(s) = 1 | + (0.366 + 1.36i)2-s + (−3.15 + 1.82i)3-s + (−1.73 + i)4-s + (−0.866 + 3.23i)5-s + (−3.64 − 3.64i)6-s + (−5.59 − 9.69i)7-s + (−2 − 1.99i)8-s + (2.12 − 3.68i)9-s − 4.73·10-s + 17.4i·11-s + (3.64 − 6.30i)12-s + (−5.11 + 19.1i)13-s + (11.1 − 11.1i)14-s + (−3.15 − 11.7i)15-s + (1.99 − 3.46i)16-s + (16.2 − 4.34i)17-s + ⋯ |
L(s) = 1 | + (0.183 + 0.683i)2-s + (−1.05 + 0.606i)3-s + (−0.433 + 0.250i)4-s + (−0.173 + 0.646i)5-s + (−0.606 − 0.606i)6-s + (−0.799 − 1.38i)7-s + (−0.250 − 0.249i)8-s + (0.236 − 0.408i)9-s − 0.473·10-s + 1.59i·11-s + (0.303 − 0.525i)12-s + (−0.393 + 1.46i)13-s + (0.799 − 0.799i)14-s + (−0.210 − 0.784i)15-s + (0.124 − 0.216i)16-s + (0.953 − 0.255i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0509i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0154398 + 0.605541i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0154398 + 0.605541i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 37 | \( 1 + (10.2 - 35.5i)T \) |
good | 3 | \( 1 + (3.15 - 1.82i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (0.866 - 3.23i)T + (-21.6 - 12.5i)T^{2} \) |
| 7 | \( 1 + (5.59 + 9.69i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 17.4iT - 121T^{2} \) |
| 13 | \( 1 + (5.11 - 19.1i)T + (-146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (-16.2 + 4.34i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-1.67 + 6.25i)T + (-312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (-2.09 - 2.09i)T + 529iT^{2} \) |
| 29 | \( 1 + (28.8 - 28.8i)T - 841iT^{2} \) |
| 31 | \( 1 + (4.29 - 4.29i)T - 961iT^{2} \) |
| 41 | \( 1 + (-42.0 + 24.2i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (22.4 + 22.4i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 13.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-41.2 + 71.5i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-4.02 + 1.07i)T + (3.01e3 - 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-23.2 - 6.22i)T + (3.22e3 + 1.86e3i)T^{2} \) |
| 67 | \( 1 + (12.5 - 7.23i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-54.7 - 94.8i)T + (-2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + 47.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (39.2 - 146. i)T + (-5.40e3 - 3.12e3i)T^{2} \) |
| 83 | \( 1 + (-16.4 + 28.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (12.0 + 45.0i)T + (-6.85e3 + 3.96e3i)T^{2} \) |
| 97 | \( 1 + (50.0 + 50.0i)T + 9.40e3iT^{2} \) |
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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
−14.91303955484384026806871160897, −14.02034714700084278178347276868, −12.69777955338275398450855028115, −11.49561294533825432151775822108, −10.30501350732118148704224743446, −9.578503024773843752824981588175, −7.19687178039513793299035367873, −6.83557462880079706884859198189, −5.04819526633100799745261451548, −3.91834189605795842790369683857,
0.56107691336599234447883931808, 3.11398359967208730135267676266, 5.56999655059796748224749363126, 5.89801864592877925215464504539, 8.127433640208280479134166803398, 9.329193120890779606897583142770, 10.77424513160067610665383311119, 11.88461757181011438268827371215, 12.51550022726119648253194024384, 13.16514372684763714747347874873