L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−0.339 − 0.0908i)5-s + (2.97 + 0.797i)7-s + (−0.707 − 0.707i)8-s + (−0.304 + 0.175i)10-s + (2.35 + 2.35i)11-s + (2.60 − 2.49i)13-s + (2.66 − 1.54i)14-s − 1.00·16-s + (−3.87 + 6.71i)17-s + (3.67 − 0.984i)19-s + (−0.0908 + 0.339i)20-s + 3.32·22-s + (3.34 − 5.78i)23-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.151 − 0.0406i)5-s + (1.12 + 0.301i)7-s + (−0.250 − 0.250i)8-s + (−0.0961 + 0.0555i)10-s + (0.709 + 0.709i)11-s + (0.723 − 0.690i)13-s + (0.713 − 0.411i)14-s − 0.250·16-s + (−0.940 + 1.62i)17-s + (0.842 − 0.225i)19-s + (−0.0203 + 0.0758i)20-s + 0.709·22-s + (0.696 − 1.20i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.750 + 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.09599 - 0.792034i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.09599 - 0.792034i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-2.60 + 2.49i)T \) |
good | 5 | \( 1 + (0.339 + 0.0908i)T + (4.33 + 2.5i)T^{2} \) |
| 7 | \( 1 + (-2.97 - 0.797i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.35 - 2.35i)T + 11iT^{2} \) |
| 17 | \( 1 + (3.87 - 6.71i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.67 + 0.984i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-3.34 + 5.78i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.68iT - 29T^{2} \) |
| 31 | \( 1 + (-0.293 + 1.09i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (-8.90 - 2.38i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.678 - 2.53i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (3.68 - 2.12i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.17 + 1.11i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + (7.75 + 7.75i)T + 59iT^{2} \) |
| 61 | \( 1 + (-5.01 - 8.69i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (13.3 - 3.56i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-0.343 - 1.28i)T + (-61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (2.19 - 2.19i)T - 73iT^{2} \) |
| 79 | \( 1 + (6.22 - 10.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.05 - 3.92i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (1.43 - 5.37i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (0.218 - 0.816i)T + (-84.0 - 48.5i)T^{2} \) |
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
−10.55831090109355857556813434476, −9.626475466704547844596473613027, −8.539680443564244390660695187294, −7.952994516680609186358433088111, −6.59551975679610779493854504999, −5.78064442949038547878190102044, −4.59873381229569402728677702024, −4.00970281979773894905365489664, −2.49846309281504426862619964546, −1.34848634599719513861822936197,
1.42229370716947420612601577356, 3.15827227264436578004411660000, 4.20685791281675619221676282006, 5.07516997993696705601487679602, 6.02913561028382573797481976312, 7.16004758246433146748663721113, 7.66252207929772145850122250251, 8.880789722193051318450858162638, 9.313644688956026752286220197629, 10.94884838004601113201340970757