L(s) = 1 | + (0.471 − 0.881i)2-s + (−0.555 − 0.831i)4-s + (0.0980 − 0.995i)7-s + (−0.995 + 0.0980i)8-s + (0.634 − 0.773i)9-s + (−0.0238 + 0.485i)11-s + (−0.831 − 0.555i)14-s + (−0.382 + 0.923i)16-s + (−0.382 − 0.923i)18-s + (0.416 + 0.249i)22-s + (−0.728 − 1.36i)23-s + (−0.956 + 0.290i)25-s + (−0.881 + 0.471i)28-s + (1.32 − 1.46i)29-s + (0.634 + 0.773i)32-s + ⋯ |
L(s) = 1 | + (0.471 − 0.881i)2-s + (−0.555 − 0.831i)4-s + (0.0980 − 0.995i)7-s + (−0.995 + 0.0980i)8-s + (0.634 − 0.773i)9-s + (−0.0238 + 0.485i)11-s + (−0.831 − 0.555i)14-s + (−0.382 + 0.923i)16-s + (−0.382 − 0.923i)18-s + (0.416 + 0.249i)22-s + (−0.728 − 1.36i)23-s + (−0.956 + 0.290i)25-s + (−0.881 + 0.471i)28-s + (1.32 − 1.46i)29-s + (0.634 + 0.773i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.303461373\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.303461373\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.471 + 0.881i)T \) |
| 7 | \( 1 + (-0.0980 + 0.995i)T \) |
good | 3 | \( 1 + (-0.634 + 0.773i)T^{2} \) |
| 5 | \( 1 + (0.956 - 0.290i)T^{2} \) |
| 11 | \( 1 + (0.0238 - 0.485i)T + (-0.995 - 0.0980i)T^{2} \) |
| 13 | \( 1 + (0.290 - 0.956i)T^{2} \) |
| 17 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 19 | \( 1 + (0.881 - 0.471i)T^{2} \) |
| 23 | \( 1 + (0.728 + 1.36i)T + (-0.555 + 0.831i)T^{2} \) |
| 29 | \( 1 + (-1.32 + 1.46i)T + (-0.0980 - 0.995i)T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.0504 - 0.0841i)T + (-0.471 + 0.881i)T^{2} \) |
| 41 | \( 1 + (-0.831 - 0.555i)T^{2} \) |
| 43 | \( 1 + (0.633 - 1.33i)T + (-0.634 - 0.773i)T^{2} \) |
| 47 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 53 | \( 1 + (-1.26 - 1.39i)T + (-0.0980 + 0.995i)T^{2} \) |
| 59 | \( 1 + (0.290 + 0.956i)T^{2} \) |
| 61 | \( 1 + (0.773 + 0.634i)T^{2} \) |
| 67 | \( 1 + (0.805 + 0.288i)T + (0.773 + 0.634i)T^{2} \) |
| 71 | \( 1 + (-0.301 + 0.247i)T + (0.195 - 0.980i)T^{2} \) |
| 73 | \( 1 + (0.980 - 0.195i)T^{2} \) |
| 79 | \( 1 + (-0.113 + 0.569i)T + (-0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (0.471 + 0.881i)T^{2} \) |
| 89 | \( 1 + (0.555 + 0.831i)T^{2} \) |
| 97 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
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
−9.557570056640291054066143615319, −8.493629524779631927267466353758, −7.58748717650147223994580455971, −6.58414805329347771051174930455, −5.98562663589871429723258555280, −4.48347882530753672422103689553, −4.35670802343367072263063092762, −3.28440573487468933344923331038, −2.09790960382746045869135236633, −0.879490321285036588000163682783,
1.96631542608826617629716128767, 3.16268793168900885070547333671, 4.12350121849901264566584191723, 5.18775435097005200629370575759, 5.58856443221051770736516915086, 6.57164078122298421295102952652, 7.36510944462419062486255163552, 8.193087705912769708226919518816, 8.681718415170750627896242878179, 9.611456662091780465865658513352