| 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
−9.675469999657216636874090023670, −8.712572433673920965502637146025, −8.364233178785128344398035808985, −7.23584030757861214270040522074, −6.68047208665476241283162642064, −5.16645523419492230492081799261, −4.86012359757556361137111673558, −3.92716459948879044681183296149, −2.47738690578755880020146647080, −1.40983686235467803294787518095,
0.45067139771728568620706867685, 2.20359469163342630742261339803, 3.33949032735537638685494461587, 3.99630645893780388266443603779, 5.22115381496844562546035815828, 6.16255170325804459690583812212, 6.91581532993709999893223995020, 7.74275847081371917928485496632, 8.451930663381957291601269424263, 9.265891608113872170054977934483
