L(s) = 1 | − 2-s + i·3-s + 4-s − i·6-s − 5.12·7-s − 8-s − 9-s − 3.12i·11-s + i·12-s + (−3.56 − 0.561i)13-s + 5.12·14-s + 16-s − 2i·17-s + 18-s + 6i·19-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577i·3-s + 0.5·4-s − 0.408i·6-s − 1.93·7-s − 0.353·8-s − 0.333·9-s − 0.941i·11-s + 0.288i·12-s + (−0.987 − 0.155i)13-s + 1.36·14-s + 0.250·16-s − 0.485i·17-s + 0.235·18-s + 1.37i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.953 - 0.302i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6425622787\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6425622787\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (3.56 + 0.561i)T \) |
good | 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 + 3.12iT - 11T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 6iT - 19T^{2} \) |
| 23 | \( 1 - 3.12iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 5.12iT - 31T^{2} \) |
| 37 | \( 1 - 3.12T + 37T^{2} \) |
| 41 | \( 1 - 9.12iT - 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 - 11.3iT - 53T^{2} \) |
| 59 | \( 1 + 7.12iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 6.24iT - 71T^{2} \) |
| 73 | \( 1 - 4.87T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 - 5.12iT - 89T^{2} \) |
| 97 | \( 1 + 4.87T + 97T^{2} \) |
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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
−9.575422422388446253970176143250, −8.629152607758265023674191355330, −7.70812704497642208237360902455, −6.93462406621761804570347254689, −5.99888884514053454801064785707, −5.54730436924419985375046819052, −3.99767581267546126637115152324, −3.27905812853520514898777179710, −2.46502565400019801682730491767, −0.54591805223164192857838228813,
0.57908695306657818107987516020, 2.24892274606186142265257280938, 2.86503232580270665550045657231, 4.06589803563988310379639525664, 5.32076488195697486343266576098, 6.35405805250604899403397625377, 7.00028026340007679868836982842, 7.26416552012837018225694150819, 8.509714090634218514944356124121, 9.224422100657512587785066072919