L(s) = 1 | + (0.0461 − 0.141i)3-s + (0.383 − 2.20i)5-s − 3.58·7-s + (2.40 + 1.75i)9-s + (4.77 − 3.46i)11-s + (−3.77 − 2.74i)13-s + (−0.295 − 0.156i)15-s + (−1.19 − 3.67i)17-s + (−0.680 − 2.09i)19-s + (−0.165 + 0.509i)21-s + (2.97 − 2.16i)23-s + (−4.70 − 1.69i)25-s + (0.721 − 0.524i)27-s + (−0.982 + 3.02i)29-s + (−1.10 − 3.39i)31-s + ⋯ |
L(s) = 1 | + (0.0266 − 0.0819i)3-s + (0.171 − 0.985i)5-s − 1.35·7-s + (0.803 + 0.583i)9-s + (1.43 − 1.04i)11-s + (−1.04 − 0.760i)13-s + (−0.0761 − 0.0402i)15-s + (−0.289 − 0.890i)17-s + (−0.156 − 0.480i)19-s + (−0.0361 + 0.111i)21-s + (0.620 − 0.450i)23-s + (−0.941 − 0.338i)25-s + (0.138 − 0.100i)27-s + (−0.182 + 0.561i)29-s + (−0.198 − 0.609i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0948 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0948 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.900764 - 0.819035i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.900764 - 0.819035i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.383 + 2.20i)T \) |
good | 3 | \( 1 + (-0.0461 + 0.141i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + 3.58T + 7T^{2} \) |
| 11 | \( 1 + (-4.77 + 3.46i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (3.77 + 2.74i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (1.19 + 3.67i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.680 + 2.09i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.97 + 2.16i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.982 - 3.02i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.10 + 3.39i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-8.01 - 5.82i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (1.74 + 1.26i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 3.33T + 43T^{2} \) |
| 47 | \( 1 + (1.74 - 5.37i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.830 + 2.55i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.72 - 2.70i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.08 + 2.23i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-2.97 - 9.16i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (2.08 - 6.42i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.72 + 4.88i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.83 - 8.72i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.500 - 1.54i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (5.59 - 4.06i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (3.70 - 11.4i)T + (-78.4 - 57.0i)T^{2} \) |
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
−11.11171068855446382021890095085, −9.808996830373429208267950311759, −9.418009939865029344166925956049, −8.446984676784799221502400563241, −7.21091481989112952901342274838, −6.35712318709973740757671598884, −5.19972758492546390383389547841, −4.11831831162973258570600116987, −2.75207569602248002702823967540, −0.826512997043015275928671102059,
1.99080681393859835604428320973, 3.51537242264570302436303846127, 4.28875529383788891073288300020, 6.16320765933426447645592628349, 6.77622742330932805839492425408, 7.34782991255542477549382893844, 9.232508679189966772520539559234, 9.635336295831580563157851435679, 10.31126671265910555731330490790, 11.57124278377304254800609877914