L(s) = 1 | − 2.23i·5-s − 3.40·7-s + 8.20i·17-s + 4.79i·23-s − 5.00·25-s − 10.0i·29-s + 7.95·31-s + 7.60i·35-s + 1.73·37-s − 3.98i·41-s + 13.1·43-s + 4.56·49-s + 11.3i·53-s − 13.2i·59-s + 8.68·67-s + ⋯ |
L(s) = 1 | − 0.999i·5-s − 1.28·7-s + 1.99i·17-s + 0.999i·23-s − 1.00·25-s − 1.87i·29-s + 1.42·31-s + 1.28i·35-s + 0.285·37-s − 0.622i·41-s + 1.99·43-s + 0.652·49-s + 1.55i·53-s − 1.71i·59-s + 1.06·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.339829757\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.339829757\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 - 4.79iT \) |
good | 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 8.20iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 29 | \( 1 + 10.0iT - 29T^{2} \) |
| 31 | \( 1 - 7.95T + 31T^{2} \) |
| 37 | \( 1 - 1.73T + 37T^{2} \) |
| 41 | \( 1 + 3.98iT - 41T^{2} \) |
| 43 | \( 1 - 13.1T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 11.3iT - 53T^{2} \) |
| 59 | \( 1 + 13.2iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 8.68T + 67T^{2} \) |
| 71 | \( 1 + 6.99iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 6.86iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−8.151298560658309643957067418332, −7.87644226502184143668619155735, −6.66054023279208378709262040324, −6.02882720253262315774383430730, −5.55839029100104378630781708832, −4.26542833816540306246457515426, −3.93332651098257452063488596730, −2.82221471335863771958746265100, −1.72294990645388852377210066209, −0.54019843943366028455361021002,
0.78326786043994733520547785553, 2.61929613754241521817641031516, 2.83552163017422209590146759292, 3.78946332702303618907502675141, 4.78114170639828505102748324576, 5.69509012824152453337940472274, 6.59357942368613436347270811265, 6.88477822655428578303571668857, 7.58540667026737219572635613829, 8.603959949757752431727831027104