L(s) = 1 | + (0.881 + 0.471i)2-s + (0.555 + 0.831i)4-s + (0.995 + 0.0980i)7-s + (0.0980 + 0.995i)8-s + (−0.773 − 0.634i)9-s + (1.27 − 1.15i)11-s + (0.831 + 0.555i)14-s + (−0.382 + 0.923i)16-s + (−0.382 − 0.923i)18-s + (1.66 − 0.416i)22-s + (−1.11 + 0.598i)23-s + (−0.290 − 0.956i)25-s + (0.471 + 0.881i)28-s + (0.0788 + 1.60i)29-s + (−0.773 + 0.634i)32-s + ⋯ |
L(s) = 1 | + (0.881 + 0.471i)2-s + (0.555 + 0.831i)4-s + (0.995 + 0.0980i)7-s + (0.0980 + 0.995i)8-s + (−0.773 − 0.634i)9-s + (1.27 − 1.15i)11-s + (0.831 + 0.555i)14-s + (−0.382 + 0.923i)16-s + (−0.382 − 0.923i)18-s + (1.66 − 0.416i)22-s + (−1.11 + 0.598i)23-s + (−0.290 − 0.956i)25-s + (0.471 + 0.881i)28-s + (0.0788 + 1.60i)29-s + (−0.773 + 0.634i)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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.074040164\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.074040164\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.881 - 0.471i)T \) |
| 7 | \( 1 + (-0.995 - 0.0980i)T \) |
good | 3 | \( 1 + (0.773 + 0.634i)T^{2} \) |
| 5 | \( 1 + (0.290 + 0.956i)T^{2} \) |
| 11 | \( 1 + (-1.27 + 1.15i)T + (0.0980 - 0.995i)T^{2} \) |
| 13 | \( 1 + (-0.956 - 0.290i)T^{2} \) |
| 17 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 19 | \( 1 + (-0.471 - 0.881i)T^{2} \) |
| 23 | \( 1 + (1.11 - 0.598i)T + (0.555 - 0.831i)T^{2} \) |
| 29 | \( 1 + (-0.0788 - 1.60i)T + (-0.995 + 0.0980i)T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (-0.360 + 1.43i)T + (-0.881 - 0.471i)T^{2} \) |
| 41 | \( 1 + (0.831 + 0.555i)T^{2} \) |
| 43 | \( 1 + (1.88 - 0.672i)T + (0.773 - 0.634i)T^{2} \) |
| 47 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 53 | \( 1 + (0.0887 - 1.80i)T + (-0.995 - 0.0980i)T^{2} \) |
| 59 | \( 1 + (-0.956 + 0.290i)T^{2} \) |
| 61 | \( 1 + (0.634 - 0.773i)T^{2} \) |
| 67 | \( 1 + (1.70 - 0.805i)T + (0.634 - 0.773i)T^{2} \) |
| 71 | \( 1 + (0.247 + 0.301i)T + (-0.195 + 0.980i)T^{2} \) |
| 73 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 79 | \( 1 + (-0.373 + 1.87i)T + (-0.923 - 0.382i)T^{2} \) |
| 83 | \( 1 + (0.881 - 0.471i)T^{2} \) |
| 89 | \( 1 + (-0.555 - 0.831i)T^{2} \) |
| 97 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.196114786719615629744197291818, −8.633090384274284178264840399064, −7.996135314285183176026402812226, −7.04243887484999332513701466125, −6.05995622500749312134377904163, −5.76438560231111184710615933492, −4.61798945991829713586226224983, −3.78937190016911049208239314679, −2.99832049013624461887947935216, −1.61275917254156756834825928962,
1.61061764082021150695035890797, 2.28559824886715064869792801154, 3.63644787205720967623307341980, 4.47997798109680190859908731523, 5.07110934710757056676203834440, 6.04727942269272832327624831086, 6.82229391928805315504139886849, 7.77416512185137709131543568740, 8.538730733302974213099320629397, 9.677682375087689058818392668389