L(s) = 1 | + (−0.965 − 0.258i)3-s + (0.866 + 0.499i)9-s − 1.73i·11-s + (−0.707 − 0.707i)17-s + i·19-s + (−0.707 − 0.707i)27-s + (−0.448 + 1.67i)33-s − 1.73i·41-s − i·49-s + (0.500 + 0.866i)51-s + (0.258 − 0.965i)57-s + (1.22 − 1.22i)67-s + (−1.22 − 1.22i)73-s + (0.500 + 0.866i)81-s + (0.707 − 0.707i)83-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)3-s + (0.866 + 0.499i)9-s − 1.73i·11-s + (−0.707 − 0.707i)17-s + i·19-s + (−0.707 − 0.707i)27-s + (−0.448 + 1.67i)33-s − 1.73i·41-s − i·49-s + (0.500 + 0.866i)51-s + (0.258 − 0.965i)57-s + (1.22 − 1.22i)67-s + (−1.22 − 1.22i)73-s + (0.500 + 0.866i)81-s + (0.707 − 0.707i)83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6939123010\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6939123010\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 19 | \( 1 - iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + 1.73iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.22 + 1.22i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 89 | \( 1 + 1.73T + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−8.859321143022208533515316604336, −8.168594799834553327118113047838, −7.29310743767258455095823317365, −6.50902046730052476828820007928, −5.80343440588807418951183394677, −5.23292024261843342587094046490, −4.16510085485276716052308196890, −3.24020540286045557931749766086, −1.92069277720190121828216721513, −0.55239926231527711127773673288,
1.43297833105325966581701792241, 2.58855474464376291379027404222, 4.07388350991801183180656613757, 4.58095317800175320138707220929, 5.33188135867451845002843958128, 6.36794896274370452350263276978, 6.90009740200966599373718489331, 7.62455726435207694177103418558, 8.697345513498091414497282235694, 9.612658540091526424584382052585