L(s) = 1 | + (−0.437 + 1.67i)3-s + (−1.41 + 2.23i)7-s + (−2.61 − 1.46i)9-s + 3.83i·11-s + 5.99·13-s − 2.07i·17-s + 5.11i·19-s + (−3.12 − 3.34i)21-s + 8.57·23-s + (3.59 − 3.74i)27-s + 5.64i·29-s + 1.95i·31-s + (−6.42 − 1.67i)33-s + 2.23i·37-s + (−2.61 + 10.0i)39-s + ⋯ |
L(s) = 1 | + (−0.252 + 0.967i)3-s + (−0.534 + 0.845i)7-s + (−0.872 − 0.488i)9-s + 1.15i·11-s + 1.66·13-s − 0.502i·17-s + 1.17i·19-s + (−0.682 − 0.730i)21-s + 1.78·23-s + (0.692 − 0.721i)27-s + 1.04i·29-s + 0.351i·31-s + (−1.11 − 0.291i)33-s + 0.367i·37-s + (−0.419 + 1.60i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.277688653\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277688653\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.437 - 1.67i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.41 - 2.23i)T \) |
good | 11 | \( 1 - 3.83iT - 11T^{2} \) |
| 13 | \( 1 - 5.99T + 13T^{2} \) |
| 17 | \( 1 + 2.07iT - 17T^{2} \) |
| 19 | \( 1 - 5.11iT - 19T^{2} \) |
| 23 | \( 1 - 8.57T + 23T^{2} \) |
| 29 | \( 1 - 5.64iT - 29T^{2} \) |
| 31 | \( 1 - 1.95iT - 31T^{2} \) |
| 37 | \( 1 - 2.23iT - 37T^{2} \) |
| 41 | \( 1 + 7.49T + 41T^{2} \) |
| 43 | \( 1 - 9.47iT - 43T^{2} \) |
| 47 | \( 1 + 12.9iT - 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 + 4.37iT - 61T^{2} \) |
| 67 | \( 1 - 6.70iT - 67T^{2} \) |
| 71 | \( 1 + 2.02iT - 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 7.47T + 79T^{2} \) |
| 83 | \( 1 - 7.49iT - 83T^{2} \) |
| 89 | \( 1 + 2.86T + 89T^{2} \) |
| 97 | \( 1 + 0.206T + 97T^{2} \) |
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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
−9.420689256273425891506743915144, −8.859783686276120575824832458114, −8.166498312041327679957091344844, −6.86918557689631191692809703011, −6.26429192128432559930157747313, −5.34017090387549324948579856752, −4.75059118755068781804065055320, −3.57801706121469171764952798012, −3.02927084281453845768022420025, −1.52005351641915515026291037903,
0.51212319758058525101640820975, 1.36805373955684371342528493067, 2.90711333471761733182010119463, 3.57836061804865783374845472303, 4.78222133301327048634786891940, 5.96108995628910769989679253956, 6.33674351678973558151115082752, 7.12404317785967970645019689107, 7.937343753742741523778944213616, 8.700109100886680622509125012291