L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.707 − 1.22i)3-s + (−0.499 − 0.866i)4-s + (−2.12 + 3.67i)5-s + 1.41·6-s + 0.999·8-s + (0.500 − 0.866i)9-s + (−2.12 − 3.67i)10-s + (0.5 + 0.866i)11-s + (−0.707 + 1.22i)12-s + 6·15-s + (−0.5 + 0.866i)16-s + (2.82 + 4.89i)17-s + (0.499 + 0.866i)18-s + 4.24·20-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.408 − 0.707i)3-s + (−0.249 − 0.433i)4-s + (−0.948 + 1.64i)5-s + 0.577·6-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.670 − 1.16i)10-s + (0.150 + 0.261i)11-s + (−0.204 + 0.353i)12-s + 1.54·15-s + (−0.125 + 0.216i)16-s + (0.685 + 1.18i)17-s + (0.117 + 0.204i)18-s + 0.948·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.435i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2256999079\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2256999079\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 3 | \( 1 + (0.707 + 1.22i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (2.12 - 3.67i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-2.82 - 4.89i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-0.707 - 1.22i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (2.12 - 3.67i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4 + 6.92i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.707 + 1.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.41 + 2.44i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (4.24 + 7.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.9T + 83T^{2} \) |
| 89 | \( 1 + (-3.53 + 6.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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
−10.27808716957043006502343809939, −9.697047007462055140865701942212, −8.265543480491468488375416143855, −7.71643245441955268406703975218, −6.96075729797655424594791704208, −6.47817717615397359755867292861, −5.66249895702367805770359502572, −4.08065735938503601543205085087, −3.30474698474347804759164360260, −1.70242775264785880087716475833,
0.12877449719525281497634525229, 1.39861236430928194746645600693, 3.16051524306858387648923877109, 4.34360222680365389620298447198, 4.70911807006801719758268150721, 5.58745505324211400281561294195, 7.14514558720728711668102465556, 8.218281649469284335380126380884, 8.496412942317778106043445332457, 9.622932264419643056374448186191