L(s) = 1 | + (−0.809 − 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.951i)6-s + 4.63·7-s + (0.309 − 0.951i)8-s + (−0.809 + 0.587i)9-s + (2.05 + 1.49i)11-s + (−0.809 + 0.587i)12-s + (0.116 − 0.0846i)13-s + (−3.74 − 2.72i)14-s + (−0.809 + 0.587i)16-s + (2.31 − 7.12i)17-s + 0.999·18-s + (−2.08 + 6.41i)19-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.126 − 0.388i)6-s + 1.75·7-s + (0.109 − 0.336i)8-s + (−0.269 + 0.195i)9-s + (0.619 + 0.450i)11-s + (−0.233 + 0.169i)12-s + (0.0323 − 0.0234i)13-s + (−1.00 − 0.727i)14-s + (−0.202 + 0.146i)16-s + (0.561 − 1.72i)17-s + 0.235·18-s + (−0.477 + 1.47i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.51029 + 0.185756i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.51029 + 0.185756i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4.63T + 7T^{2} \) |
| 11 | \( 1 + (-2.05 - 1.49i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.116 + 0.0846i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.31 + 7.12i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.08 - 6.41i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.35 + 0.985i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.696 + 2.14i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.310 + 0.954i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-0.0719 + 0.0523i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (2.48 - 1.80i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 9.02T + 43T^{2} \) |
| 47 | \( 1 + (-3.36 - 10.3i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.53 + 4.72i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (4.25 - 3.08i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-11.0 - 8.05i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (2.36 - 7.27i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (3.17 + 9.76i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.59 + 1.15i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.230 - 0.710i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.958 - 2.95i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (0.593 + 0.431i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.81 + 8.67i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.32572934727442351467110955669, −9.590116828581778191650850971821, −8.749506279664822466452823283557, −7.930764159355014591735181509612, −7.33615087447482723720433948847, −5.81641179230925406849907532461, −4.73346551023644984311910485098, −3.98838791655549327725069947319, −2.52409731690234515108136569963, −1.35484677019227857502566865287,
1.17562056151172202126716056166, 2.15007055811785485995582882279, 3.92098832794541791185736847375, 5.09456431107598817984373208103, 6.03665954623394064257228237423, 7.00844907038586633780354238319, 7.889692711407244470536777725445, 8.499802766556492857053196274339, 9.073142906064097104106268078731, 10.45450728259707061552804611418