L(s) = 1 | + i·2-s − 3-s − 4-s + i·5-s − i·6-s − 3.04i·7-s − i·8-s + 9-s − 10-s + 6.24i·11-s + 12-s + 3.04·14-s − i·15-s + 16-s − 2.69·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577·3-s − 0.5·4-s + 0.447i·5-s − 0.408i·6-s − 1.15i·7-s − 0.353i·8-s + 0.333·9-s − 0.316·10-s + 1.88i·11-s + 0.288·12-s + 0.814·14-s − 0.258i·15-s + 0.250·16-s − 0.652·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4659119385\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4659119385\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 + 3.04iT - 7T^{2} \) |
| 11 | \( 1 - 6.24iT - 11T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 + 5.82iT - 19T^{2} \) |
| 23 | \( 1 - 5.62T + 23T^{2} \) |
| 29 | \( 1 + 5.14T + 29T^{2} \) |
| 31 | \( 1 + 3.53iT - 31T^{2} \) |
| 37 | \( 1 - 4.65iT - 37T^{2} \) |
| 41 | \( 1 + 3.77iT - 41T^{2} \) |
| 43 | \( 1 + 2.85T + 43T^{2} \) |
| 47 | \( 1 - 3.61iT - 47T^{2} \) |
| 53 | \( 1 - 0.664T + 53T^{2} \) |
| 59 | \( 1 - 12.9iT - 59T^{2} \) |
| 61 | \( 1 + 7.91T + 61T^{2} \) |
| 67 | \( 1 + 0.198iT - 67T^{2} \) |
| 71 | \( 1 + 0.374iT - 71T^{2} \) |
| 73 | \( 1 + 14.6iT - 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 - 6.83iT - 83T^{2} \) |
| 89 | \( 1 + 9.50iT - 89T^{2} \) |
| 97 | \( 1 + 0.335iT - 97T^{2} \) |
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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
−7.54887138800022430151989687660, −7.29429530488744091308052184010, −6.84924550155221118165209126975, −6.11466367235236404941395932941, −4.98288025556001266193602737423, −4.59026340883535608557485779024, −3.90860469256281205758604388599, −2.67124140197049916935529837607, −1.47793356669188518486648387051, −0.15617528345317018793389955649,
1.03768067403991475902943316255, 2.02759929855120867056044307394, 3.08865171667360501394372639896, 3.73408724790913424148030118897, 4.79702321718466380241873742006, 5.60785214790020570245064123189, 5.82921030463739087310207151234, 6.78172099586468682897149040085, 7.957824928330762197731059875826, 8.583319393888445834594032848219