L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.988 + 0.718i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)5-s + (0.377 + 1.16i)6-s + (0.809 + 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.465 + 1.43i)9-s − 0.999·10-s + (0.912 − 3.18i)11-s + 1.22·12-s + (1.07 − 3.30i)13-s + (0.809 − 0.587i)14-s + (0.988 + 0.718i)15-s + (0.309 + 0.951i)16-s + (−0.0426 − 0.131i)17-s + ⋯ |
L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.570 + 0.414i)3-s + (−0.404 − 0.293i)4-s + (−0.138 − 0.425i)5-s + (0.154 + 0.474i)6-s + (0.305 + 0.222i)7-s + (−0.286 + 0.207i)8-s + (−0.155 + 0.477i)9-s − 0.316·10-s + (0.275 − 0.961i)11-s + 0.352·12-s + (0.298 − 0.917i)13-s + (0.216 − 0.157i)14-s + (0.255 + 0.185i)15-s + (0.0772 + 0.237i)16-s + (−0.0103 − 0.0318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 + 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 + 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.518004 - 0.922015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.518004 - 0.922015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (0.309 + 0.951i)T \) |
| 7 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 + (-0.912 + 3.18i)T \) |
good | 3 | \( 1 + (0.988 - 0.718i)T + (0.927 - 2.85i)T^{2} \) |
| 13 | \( 1 + (-1.07 + 3.30i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.0426 + 0.131i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (0.416 - 0.302i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 1.51T + 23T^{2} \) |
| 29 | \( 1 + (7.47 + 5.43i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-3.17 + 9.76i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (1.34 + 0.980i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-9.85 + 7.16i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + (-1.46 + 1.06i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.429 + 1.32i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (7.52 + 5.46i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-4.17 - 12.8i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 1.84T + 67T^{2} \) |
| 71 | \( 1 + (1.97 + 6.07i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.95 - 6.50i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.08 + 9.49i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.681 - 2.09i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 - 1.74T + 89T^{2} \) |
| 97 | \( 1 + (1.36 - 4.19i)T + (-78.4 - 57.0i)T^{2} \) |
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
−10.23332601561377900624150968246, −9.336642783432106625865104423947, −8.411660750783145768672465577476, −7.70126361282279919932532806676, −5.96195702718650701166723787858, −5.56486691183356078082917540166, −4.52732329127100457168945005907, −3.58796953897119792338895610933, −2.24704491706243020660887805469, −0.56142577142602890964145999971,
1.53653739108378959345099883193, 3.33022078604293612375010905937, 4.41767912060930473180648286443, 5.34972820665433701902802796698, 6.57728963165216088923976321820, 6.81837199217864207185835132668, 7.74658595918182845860486875964, 8.875108978157609608491280363325, 9.582687003331928414302947373894, 10.77034781950531178946343521422