L(s) = 1 | + (1.18 + 1.26i)3-s + (1 − 1.73i)4-s + (−0.813 − 0.469i)5-s + (−0.186 + 2.99i)9-s + (−2.87 + 1.65i)11-s + (3.37 − 0.792i)12-s + (−0.372 − 1.58i)15-s + (−1.99 − 3.46i)16-s + (−1.62 + 0.939i)20-s + (−2.87 − 1.65i)23-s + (−2.05 − 3.56i)25-s + (−4.00 + 3.31i)27-s + (−3.05 + 5.29i)31-s + (−5.5 − 1.65i)33-s + (5 + 3.31i)36-s + 12.1·37-s + ⋯ |
L(s) = 1 | + (0.684 + 0.728i)3-s + (0.5 − 0.866i)4-s + (−0.363 − 0.210i)5-s + (−0.0620 + 0.998i)9-s + (−0.866 + 0.500i)11-s + (0.973 − 0.228i)12-s + (−0.0961 − 0.409i)15-s + (−0.499 − 0.866i)16-s + (−0.363 + 0.210i)20-s + (−0.598 − 0.345i)23-s + (−0.411 − 0.713i)25-s + (−0.769 + 0.638i)27-s + (−0.549 + 0.951i)31-s + (−0.957 − 0.288i)33-s + (0.833 + 0.552i)36-s + 1.99·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.20739 + 0.0679551i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20739 + 0.0679551i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.18 - 1.26i)T \) |
| 11 | \( 1 + (2.87 - 1.65i)T \) |
good | 2 | \( 1 + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.813 + 0.469i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (2.87 + 1.65i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.05 - 5.29i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 12.1T + 37T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-11.8 + 6.85i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 11.8iT - 53T^{2} \) |
| 59 | \( 1 + (-12.6 - 7.32i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.55 - 13.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 10.8iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 16.5iT - 89T^{2} \) |
| 97 | \( 1 + (0.0584 + 0.101i)T + (-48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.23096617009642410908557449294, −13.01767498681225041307164392941, −11.60675812704474031125500806808, −10.47341347487820387673598975424, −9.820656808171214247429748722585, −8.497247340343113472245905424425, −7.31394114624828432805182236509, −5.60928290974810850798941750951, −4.35755257875907660958260563927, −2.45479558764738213060773335021,
2.50649730523536048982707790685, 3.76428668241376040415798472380, 6.11549310294296475393631446842, 7.53587474141776881299877334320, 7.959651358483919217713316062134, 9.280744764461936783193494757493, 10.97256984907390893482249688000, 11.90584862539892331619490590227, 12.92268541914789442581456543554, 13.63046764088606868027637931815