L(s) = 1 | + (0.951 − 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 − 0.587i)4-s + (−2.13 − 0.693i)5-s + (0.309 − 0.951i)6-s + (−2.54 − 0.709i)7-s + (0.587 − 0.809i)8-s + (−0.309 − 0.951i)9-s − 2.24·10-s + (1.68 − 2.85i)11-s − 0.999i·12-s + (−0.720 − 2.21i)13-s + (−2.64 + 0.112i)14-s + (−1.81 + 1.31i)15-s + (0.309 − 0.951i)16-s + (0.988 − 3.04i)17-s + ⋯ |
L(s) = 1 | + (0.672 − 0.218i)2-s + (0.339 − 0.467i)3-s + (0.404 − 0.293i)4-s + (−0.953 − 0.309i)5-s + (0.126 − 0.388i)6-s + (−0.963 − 0.268i)7-s + (0.207 − 0.286i)8-s + (−0.103 − 0.317i)9-s − 0.709·10-s + (0.508 − 0.861i)11-s − 0.288i·12-s + (−0.199 − 0.614i)13-s + (−0.706 + 0.0300i)14-s + (−0.468 + 0.340i)15-s + (0.0772 − 0.237i)16-s + (0.239 − 0.738i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.868296 - 1.41288i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.868296 - 1.41288i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (-0.587 + 0.809i)T \) |
| 7 | \( 1 + (2.54 + 0.709i)T \) |
| 11 | \( 1 + (-1.68 + 2.85i)T \) |
good | 5 | \( 1 + (2.13 + 0.693i)T + (4.04 + 2.93i)T^{2} \) |
| 13 | \( 1 + (0.720 + 2.21i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.988 + 3.04i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-0.0237 - 0.0172i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 3.27T + 23T^{2} \) |
| 29 | \( 1 + (-4.01 - 5.52i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (1.88 - 0.612i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (2.68 - 1.94i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-8.83 - 6.42i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 7.48iT - 43T^{2} \) |
| 47 | \( 1 + (-0.343 + 0.473i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.39 + 7.37i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.23 - 7.21i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.44 + 13.6i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 5.48T + 67T^{2} \) |
| 71 | \( 1 + (-3.72 + 11.4i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (0.0950 - 0.0690i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (15.6 - 5.07i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.28 + 7.02i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 11.1iT - 89T^{2} \) |
| 97 | \( 1 + (-12.3 + 4.00i)T + (78.4 - 57.0i)T^{2} \) |
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
−11.05234332259933085352534348918, −9.902672003787383953218590545162, −8.907066080277001097824680299870, −7.908842183277415865872046186676, −7.00764661582959753070763713125, −6.12477121634097562728517469691, −4.84880825774482919047333702270, −3.58960357592157069850382588485, −2.95659715279402307801038014337, −0.813268816975542724987998154113,
2.46172402371390215037196670774, 3.73395496355552736164466935109, 4.24549753070767102576236560461, 5.64529653214626086173623990259, 6.79987627732185100244274953853, 7.45654793479963653838597232881, 8.640968026775700748974504368861, 9.548989346306007953644079822510, 10.47783639457689406711122895240, 11.52778425418067018894695452328