| L(s) = 1 | + (−0.707 + 0.707i)3-s + (−2.84 − 2.84i)5-s + 4.61i·7-s − 1.00i·9-s + (1.08 + 1.08i)11-s + (3.94 − 3.94i)13-s + 4.02·15-s + 1.29·17-s + (−3.08 + 3.08i)19-s + (−3.26 − 3.26i)21-s − 4i·23-s + 11.2i·25-s + (0.707 + 0.707i)27-s + (−1.31 + 1.31i)29-s − 2.77·31-s + ⋯ |
| L(s) = 1 | + (−0.408 + 0.408i)3-s + (−1.27 − 1.27i)5-s + 1.74i·7-s − 0.333i·9-s + (0.326 + 0.326i)11-s + (1.09 − 1.09i)13-s + 1.03·15-s + 0.314·17-s + (−0.707 + 0.707i)19-s + (−0.711 − 0.711i)21-s − 0.834i·23-s + 2.24i·25-s + (0.136 + 0.136i)27-s + (−0.244 + 0.244i)29-s − 0.498·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1268100585\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1268100585\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| good | 5 | \( 1 + (2.84 + 2.84i)T + 5iT^{2} \) |
| 7 | \( 1 - 4.61iT - 7T^{2} \) |
| 11 | \( 1 + (-1.08 - 1.08i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3.94 + 3.94i)T - 13iT^{2} \) |
| 17 | \( 1 - 1.29T + 17T^{2} \) |
| 19 | \( 1 + (3.08 - 3.08i)T - 19iT^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + (1.31 - 1.31i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.77T + 31T^{2} \) |
| 37 | \( 1 + (2.81 + 2.81i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.03iT - 41T^{2} \) |
| 43 | \( 1 + (-2.14 - 2.14i)T + 43iT^{2} \) |
| 47 | \( 1 + 9.65T + 47T^{2} \) |
| 53 | \( 1 + (6.07 + 6.07i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.05 + 4.05i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.40 - 5.40i)T - 61iT^{2} \) |
| 67 | \( 1 + (6.21 - 6.21i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.88iT - 71T^{2} \) |
| 73 | \( 1 - 3.50iT - 73T^{2} \) |
| 79 | \( 1 + 8.18T + 79T^{2} \) |
| 83 | \( 1 + (2.91 - 2.91i)T - 83iT^{2} \) |
| 89 | \( 1 - 7.98iT - 89T^{2} \) |
| 97 | \( 1 + 1.40T + 97T^{2} \) |
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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
−9.662306346538040644661535622414, −8.835806336327972932283914977158, −8.436379169428853907137828555767, −7.80526662162977593670024170335, −6.35656360035748022405607011218, −5.60162170883820581987132290897, −4.94736937806823819594620537206, −4.03530789261487628520524712993, −3.13817501591652557137281525374, −1.49853935008035690298358533652,
0.05669298440852078856319161150, 1.47799557075209854885698426772, 3.23164871893845920339927978172, 3.87398320563547547193560615545, 4.53705540460901118857092229534, 6.20880388654853406122445440383, 6.75056848436954799417050001972, 7.35728451690259986284747139918, 7.905605523316134703018568712175, 8.954483060136448646507301170351
