L(s) = 1 | + (1.37 − 1.37i)3-s − 2.73i·7-s − 0.755i·9-s + (−4.12 − 4.12i)11-s + (1.37 − 1.37i)13-s + 4.94·17-s + (0.292 − 0.292i)19-s + (−3.74 − 3.74i)21-s − 1.64i·23-s + (3.07 + 3.07i)27-s + (−5.67 + 5.67i)29-s − 3.95·31-s − 11.2·33-s + (−2.48 − 2.48i)37-s − 3.77i·39-s + ⋯ |
L(s) = 1 | + (0.791 − 0.791i)3-s − 1.03i·7-s − 0.251i·9-s + (−1.24 − 1.24i)11-s + (0.382 − 0.382i)13-s + 1.20·17-s + (0.0671 − 0.0671i)19-s + (−0.817 − 0.817i)21-s − 0.343i·23-s + (0.591 + 0.591i)27-s + (−1.05 + 1.05i)29-s − 0.710·31-s − 1.96·33-s + (−0.408 − 0.408i)37-s − 0.605i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.811943389\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.811943389\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-1.37 + 1.37i)T - 3iT^{2} \) |
| 7 | \( 1 + 2.73iT - 7T^{2} \) |
| 11 | \( 1 + (4.12 + 4.12i)T + 11iT^{2} \) |
| 13 | \( 1 + (-1.37 + 1.37i)T - 13iT^{2} \) |
| 17 | \( 1 - 4.94T + 17T^{2} \) |
| 19 | \( 1 + (-0.292 + 0.292i)T - 19iT^{2} \) |
| 23 | \( 1 + 1.64iT - 23T^{2} \) |
| 29 | \( 1 + (5.67 - 5.67i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.95T + 31T^{2} \) |
| 37 | \( 1 + (2.48 + 2.48i)T + 37iT^{2} \) |
| 41 | \( 1 + 8.40iT - 41T^{2} \) |
| 43 | \( 1 + (3.22 + 3.22i)T + 43iT^{2} \) |
| 47 | \( 1 + 5.19T + 47T^{2} \) |
| 53 | \( 1 + (7.20 + 7.20i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.41 - 6.41i)T + 59iT^{2} \) |
| 61 | \( 1 + (3.82 - 3.82i)T - 61iT^{2} \) |
| 67 | \( 1 + (-5.76 + 5.76i)T - 67iT^{2} \) |
| 71 | \( 1 - 7.92iT - 71T^{2} \) |
| 73 | \( 1 + 4.36iT - 73T^{2} \) |
| 79 | \( 1 - 5.56T + 79T^{2} \) |
| 83 | \( 1 + (0.516 - 0.516i)T - 83iT^{2} \) |
| 89 | \( 1 + 6.42iT - 89T^{2} \) |
| 97 | \( 1 - 9.44T + 97T^{2} \) |
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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
−8.774658162302011273768525714708, −8.260996435576978488159911029361, −7.50930098435136281337868329274, −7.09087331510515258632775295094, −5.79485869294029414224004415972, −5.13450929486102504845117088248, −3.60199078846955758468073132236, −3.13148653673971385050905992988, −1.84975858055473398903761355323, −0.61190073927350273189478545732,
1.88076600260289549515283974264, 2.82903032381019371007252456919, 3.66551719098354202107987022735, 4.73819099327621521741891198353, 5.43056265744847248866177644897, 6.40414097630944250471372517000, 7.66241956945702132163275919422, 8.103923479503690310856034203302, 9.087589673036037892221451285263, 9.694274846095777872908738773041