L(s) = 1 | + (0.290 + 0.956i)2-s + (−0.831 + 0.555i)4-s + (−0.634 − 0.773i)7-s + (−0.773 − 0.634i)8-s + (0.995 − 0.0980i)9-s + (−0.666 − 1.86i)11-s + (0.555 − 0.831i)14-s + (0.382 − 0.923i)16-s + (0.382 + 0.923i)18-s + (1.58 − 1.17i)22-s + (−0.0569 + 0.187i)23-s + (0.471 − 0.881i)25-s + (0.956 + 0.290i)28-s + (0.929 − 0.439i)29-s + (0.995 + 0.0980i)32-s + ⋯ |
L(s) = 1 | + (0.290 + 0.956i)2-s + (−0.831 + 0.555i)4-s + (−0.634 − 0.773i)7-s + (−0.773 − 0.634i)8-s + (0.995 − 0.0980i)9-s + (−0.666 − 1.86i)11-s + (0.555 − 0.831i)14-s + (0.382 − 0.923i)16-s + (0.382 + 0.923i)18-s + (1.58 − 1.17i)22-s + (−0.0569 + 0.187i)23-s + (0.471 − 0.881i)25-s + (0.956 + 0.290i)28-s + (0.929 − 0.439i)29-s + (0.995 + 0.0980i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.037755652\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.037755652\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.290 - 0.956i)T \) |
| 7 | \( 1 + (0.634 + 0.773i)T \) |
good | 3 | \( 1 + (-0.995 + 0.0980i)T^{2} \) |
| 5 | \( 1 + (-0.471 + 0.881i)T^{2} \) |
| 11 | \( 1 + (0.666 + 1.86i)T + (-0.773 + 0.634i)T^{2} \) |
| 13 | \( 1 + (-0.881 + 0.471i)T^{2} \) |
| 17 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 19 | \( 1 + (-0.956 - 0.290i)T^{2} \) |
| 23 | \( 1 + (0.0569 - 0.187i)T + (-0.831 - 0.555i)T^{2} \) |
| 29 | \( 1 + (-0.929 + 0.439i)T + (0.634 - 0.773i)T^{2} \) |
| 31 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 37 | \( 1 + (0.401 - 0.541i)T + (-0.290 - 0.956i)T^{2} \) |
| 41 | \( 1 + (0.555 - 0.831i)T^{2} \) |
| 43 | \( 1 + (-0.0419 + 0.854i)T + (-0.995 - 0.0980i)T^{2} \) |
| 47 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 53 | \( 1 + (1.33 + 0.633i)T + (0.634 + 0.773i)T^{2} \) |
| 59 | \( 1 + (-0.881 - 0.471i)T^{2} \) |
| 61 | \( 1 + (0.0980 + 0.995i)T^{2} \) |
| 67 | \( 1 + (0.0727 + 0.0659i)T + (0.0980 + 0.995i)T^{2} \) |
| 71 | \( 1 + (0.192 - 1.95i)T + (-0.980 - 0.195i)T^{2} \) |
| 73 | \( 1 + (0.195 + 0.980i)T^{2} \) |
| 79 | \( 1 + (-1.72 - 0.344i)T + (0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (0.290 - 0.956i)T^{2} \) |
| 89 | \( 1 + (0.831 - 0.555i)T^{2} \) |
| 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
−9.350242372864031568722395230674, −8.390214414298873607861707334521, −7.911724600616986433171585907505, −6.89475586687797019108730947485, −6.42155715319076786208514060440, −5.54434609039983350441276344597, −4.57513707041333465962779675122, −3.72063604849332564179354658425, −2.95417215032123069637811295636, −0.74935667169860005449380115570,
1.62612352823070158092658829589, 2.47349395902596957110943409076, 3.46861269027200891472257526892, 4.65103403487145039881969088040, 5.00012406121765401447561371222, 6.19760713394751372929713220604, 7.08156187617012730541723809228, 7.957017717636625778357166397527, 9.140675952265432096680198494838, 9.552308244805854916520335947546