L(s) = 1 | + (0.0623 + 0.0623i)3-s + 0.375i·7-s − 2.99i·9-s + (−2.36 + 2.36i)11-s + (−1.76 − 1.76i)13-s − 4.64·17-s + (2.34 + 2.34i)19-s + (−0.0234 + 0.0234i)21-s + 2.07i·23-s + (0.373 − 0.373i)27-s + (2.55 + 2.55i)29-s − 8.51·31-s − 0.295·33-s + (−7.62 + 7.62i)37-s − 0.219i·39-s + ⋯ |
L(s) = 1 | + (0.0359 + 0.0359i)3-s + 0.142i·7-s − 0.997i·9-s + (−0.713 + 0.713i)11-s + (−0.489 − 0.489i)13-s − 1.12·17-s + (0.539 + 0.539i)19-s + (−0.00511 + 0.00511i)21-s + 0.433i·23-s + (0.0718 − 0.0718i)27-s + (0.474 + 0.474i)29-s − 1.52·31-s − 0.0513·33-s + (−1.25 + 1.25i)37-s − 0.0352i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2331099645\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2331099645\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.0623 - 0.0623i)T + 3iT^{2} \) |
| 7 | \( 1 - 0.375iT - 7T^{2} \) |
| 11 | \( 1 + (2.36 - 2.36i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.76 + 1.76i)T + 13iT^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 + (-2.34 - 2.34i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.07iT - 23T^{2} \) |
| 29 | \( 1 + (-2.55 - 2.55i)T + 29iT^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 + (7.62 - 7.62i)T - 37iT^{2} \) |
| 41 | \( 1 + 3.77iT - 41T^{2} \) |
| 43 | \( 1 + (6.21 - 6.21i)T - 43iT^{2} \) |
| 47 | \( 1 + 9.71T + 47T^{2} \) |
| 53 | \( 1 + (-3.03 + 3.03i)T - 53iT^{2} \) |
| 59 | \( 1 + (-8.11 + 8.11i)T - 59iT^{2} \) |
| 61 | \( 1 + (-0.728 - 0.728i)T + 61iT^{2} \) |
| 67 | \( 1 + (-0.969 - 0.969i)T + 67iT^{2} \) |
| 71 | \( 1 - 9.14iT - 71T^{2} \) |
| 73 | \( 1 - 7.56iT - 73T^{2} \) |
| 79 | \( 1 + 11.8T + 79T^{2} \) |
| 83 | \( 1 + (10.6 + 10.6i)T + 83iT^{2} \) |
| 89 | \( 1 + 15.7iT - 89T^{2} \) |
| 97 | \( 1 + 3.86T + 97T^{2} \) |
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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
−9.879128394452624749248023685890, −8.953284119605245410623804641718, −8.264154551301105563589317505094, −7.23235209959025072587104524728, −6.71511147180162974560744181759, −5.57360610413495857230683735023, −4.90175501314760254159281948799, −3.77878343562670076811478655830, −2.87733635248160425599147246537, −1.66180220087171328902840062760,
0.082968800523666816844939253910, 1.93545950985304496711743980901, 2.78921273820607384694403996657, 4.02807349612893815883453756764, 4.98865466659256910943523190217, 5.60653966249655943241067929961, 6.83520076414966921868373108324, 7.37748543379199076209930424267, 8.351232791114934985884863122859, 8.909682857786406760566019438955