L(s) = 1 | + 2-s + (−0.707 − 1.58i)3-s + 4-s + (1.41 − 1.73i)5-s + (−0.707 − 1.58i)6-s + 8-s + (−2.00 + 2.23i)9-s + (1.41 − 1.73i)10-s − 0.213i·11-s + (−0.707 − 1.58i)12-s − 6.70·13-s + (−3.73 − 1.01i)15-s + 16-s − 3.16i·17-s + (−2.00 + 2.23i)18-s − 4.89i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.408 − 0.912i)3-s + 0.5·4-s + (0.632 − 0.774i)5-s + (−0.288 − 0.645i)6-s + 0.353·8-s + (−0.666 + 0.745i)9-s + (0.447 − 0.547i)10-s − 0.0643i·11-s + (−0.204 − 0.456i)12-s − 1.85·13-s + (−0.965 − 0.261i)15-s + 0.250·16-s − 0.766i·17-s + (−0.471 + 0.527i)18-s − 1.12i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.702046395\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.702046395\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (0.707 + 1.58i)T \) |
| 5 | \( 1 + (-1.41 + 1.73i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 0.213iT - 11T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 17 | \( 1 + 3.16iT - 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 6.47T + 23T^{2} \) |
| 29 | \( 1 - 2.02iT - 29T^{2} \) |
| 31 | \( 1 + 0.301iT - 31T^{2} \) |
| 37 | \( 1 + 7.13iT - 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 - 2.02iT - 43T^{2} \) |
| 47 | \( 1 + 7.75iT - 47T^{2} \) |
| 53 | \( 1 - 5T + 53T^{2} \) |
| 59 | \( 1 + 4.91T + 59T^{2} \) |
| 61 | \( 1 + 3.46iT - 61T^{2} \) |
| 67 | \( 1 - 5.32iT - 67T^{2} \) |
| 71 | \( 1 + 2.02iT - 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 0.522T + 79T^{2} \) |
| 83 | \( 1 - 16.7iT - 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 3.50T + 97T^{2} \) |
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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
−9.233138621466185868951827418913, −8.148139078084367713002785114519, −7.32922908881010281618698841280, −6.71534485748204010863539529537, −5.65942739017740570276360926953, −5.16229399748725427662913617372, −4.35961563624551196427245796126, −2.64471678903771777019667879169, −2.05435750262812371804935934256, −0.50866062040120126303082961106,
2.03661580200571410342539942931, 3.00623969996457179039722433107, 4.00409791282132090479751023696, 4.80973099554493200211670268441, 5.79032189663742220101441620887, 6.20665803279453028120252192014, 7.26110159021571582478559937061, 8.128623868216300691127229709638, 9.480668755719649678828874182906, 10.01832259242993994615796209087