L(s) = 1 | + 2-s + 3-s + 4-s − 2.68·5-s + 6-s − 4.10·7-s + 8-s + 9-s − 2.68·10-s + 3.10·11-s + 12-s − 13-s − 4.10·14-s − 2.68·15-s + 16-s − 3.53·17-s + 18-s − 6.07·19-s − 2.68·20-s − 4.10·21-s + 3.10·22-s − 5.39·23-s + 24-s + 2.20·25-s − 26-s + 27-s − 4.10·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.20·5-s + 0.408·6-s − 1.55·7-s + 0.353·8-s + 0.333·9-s − 0.848·10-s + 0.937·11-s + 0.288·12-s − 0.277·13-s − 1.09·14-s − 0.692·15-s + 0.250·16-s − 0.857·17-s + 0.235·18-s − 1.39·19-s − 0.600·20-s − 0.895·21-s + 0.662·22-s − 1.12·23-s + 0.204·24-s + 0.440·25-s − 0.196·26-s + 0.192·27-s − 0.775·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.033454901\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.033454901\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 2.68T + 5T^{2} \) |
| 7 | \( 1 + 4.10T + 7T^{2} \) |
| 11 | \( 1 - 3.10T + 11T^{2} \) |
| 17 | \( 1 + 3.53T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 + 5.39T + 23T^{2} \) |
| 29 | \( 1 - 4.59T + 29T^{2} \) |
| 31 | \( 1 - 7.13T + 31T^{2} \) |
| 37 | \( 1 - 1.27T + 37T^{2} \) |
| 41 | \( 1 - 5.54T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 1.77T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 5.38T + 67T^{2} \) |
| 71 | \( 1 - 10.8T + 71T^{2} \) |
| 73 | \( 1 - 0.969T + 73T^{2} \) |
| 79 | \( 1 - 2.83T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−7.81364820818894291818222397693, −6.95403240725294488173189062307, −6.45512369840660917268096889233, −6.05032256280042690144209982853, −4.59089011163998086903954744983, −4.16750318370060673480091528583, −3.66222342511405632969083131821, −2.86748356713728642532567480477, −2.14685595131935054090951993230, −0.58352387901451015617690406329,
0.58352387901451015617690406329, 2.14685595131935054090951993230, 2.86748356713728642532567480477, 3.66222342511405632969083131821, 4.16750318370060673480091528583, 4.59089011163998086903954744983, 6.05032256280042690144209982853, 6.45512369840660917268096889233, 6.95403240725294488173189062307, 7.81364820818894291818222397693