L(s) = 1 | + (−1 + 2i)5-s + (1 − i)7-s − 6i·11-s + (−1 + i)13-s + (−1 − i)17-s − 4·19-s + (5 + 5i)23-s + (−3 − 4i)25-s − 8i·29-s − 2i·31-s + (1 + 3i)35-s + (−5 − 5i)37-s − 6·41-s + (−3 − 3i)43-s + (7 − 7i)47-s + ⋯ |
L(s) = 1 | + (−0.447 + 0.894i)5-s + (0.377 − 0.377i)7-s − 1.80i·11-s + (−0.277 + 0.277i)13-s + (−0.242 − 0.242i)17-s − 0.917·19-s + (1.04 + 1.04i)23-s + (−0.600 − 0.800i)25-s − 1.48i·29-s − 0.359i·31-s + (0.169 + 0.507i)35-s + (−0.821 − 0.821i)37-s − 0.937·41-s + (−0.457 − 0.457i)43-s + (1.02 − 1.02i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0898 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.097347260\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.097347260\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1 - 2i)T \) |
good | 7 | \( 1 + (-1 + i)T - 7iT^{2} \) |
| 11 | \( 1 + 6iT - 11T^{2} \) |
| 13 | \( 1 + (1 - i)T - 13iT^{2} \) |
| 17 | \( 1 + (1 + i)T + 17iT^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + (-5 - 5i)T + 23iT^{2} \) |
| 29 | \( 1 + 8iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + (5 + 5i)T + 37iT^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + (3 + 3i)T + 43iT^{2} \) |
| 47 | \( 1 + (-7 + 7i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1 + i)T - 53iT^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (-7 + 7i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + (-9 + 9i)T - 73iT^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (5 + 5i)T + 83iT^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + (3 + 3i)T + 97iT^{2} \) |
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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
−9.265891608113872170054977934483, −8.451930663381957291601269424263, −7.74275847081371917928485496632, −6.91581532993709999893223995020, −6.16255170325804459690583812212, −5.22115381496844562546035815828, −3.99630645893780388266443603779, −3.33949032735537638685494461587, −2.20359469163342630742261339803, −0.45067139771728568620706867685,
1.40983686235467803294787518095, 2.47738690578755880020146647080, 3.92716459948879044681183296149, 4.86012359757556361137111673558, 5.16645523419492230492081799261, 6.68047208665476241283162642064, 7.23584030757861214270040522074, 8.364233178785128344398035808985, 8.712572433673920965502637146025, 9.675469999657216636874090023670