L(s) = 1 | + (−1 − i)2-s + 2i·4-s − 3.46·5-s + (−1.73 + 2i)7-s + (2 − 2i)8-s + (3.46 + 3.46i)10-s + 2·11-s + 3.46·13-s + (3.73 − 0.267i)14-s − 4·16-s − 3.46i·17-s − 6.92i·19-s − 6.92i·20-s + (−2 − 2i)22-s + 2i·23-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + i·4-s − 1.54·5-s + (−0.654 + 0.755i)7-s + (0.707 − 0.707i)8-s + (1.09 + 1.09i)10-s + 0.603·11-s + 0.960·13-s + (0.997 − 0.0716i)14-s − 16-s − 0.840i·17-s − 1.58i·19-s − 1.54i·20-s + (−0.426 − 0.426i)22-s + 0.417i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0716 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0716 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.446445 - 0.415541i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.446445 - 0.415541i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 2iT - 23T^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 6.92iT - 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 2iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 12iT - 79T^{2} \) |
| 83 | \( 1 - 6.92iT - 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−11.03209493298645252734600025423, −9.630879303895691973540909992430, −9.010606987726053344528211334691, −8.196398448465406580953985062018, −7.32379461670124567664132011176, −6.36470307393936787382590872560, −4.58013551054993211992679579359, −3.62779921343568044316781763877, −2.67980867397175825308727844898, −0.58384853917214874937567409568,
1.11892292735081616892222788900, 3.59817122738588803860458302893, 4.26034556935623912322598867249, 5.88154691569736404913858123822, 6.76915033850979135925335320528, 7.58148818971757696102667669646, 8.342259885291102886266897157020, 9.075226766970808952043419831517, 10.42621698774774783742443779168, 10.74856997592978381163062032820