L(s) = 1 | + (−0.773 + 0.634i)2-s + (0.195 − 0.980i)4-s + (0.881 + 0.471i)7-s + (0.471 + 0.881i)8-s + (0.956 + 0.290i)9-s + (1.55 − 0.929i)11-s + (−0.980 + 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (−0.609 + 1.70i)22-s + (−0.448 − 0.368i)23-s + (−0.995 − 0.0980i)25-s + (0.634 − 0.773i)28-s + (−0.485 − 1.93i)29-s + (0.956 − 0.290i)32-s + ⋯ |
L(s) = 1 | + (−0.773 + 0.634i)2-s + (0.195 − 0.980i)4-s + (0.881 + 0.471i)7-s + (0.471 + 0.881i)8-s + (0.956 + 0.290i)9-s + (1.55 − 0.929i)11-s + (−0.980 + 0.195i)14-s + (−0.923 − 0.382i)16-s + (−0.923 + 0.382i)18-s + (−0.609 + 1.70i)22-s + (−0.448 − 0.368i)23-s + (−0.995 − 0.0980i)25-s + (0.634 − 0.773i)28-s + (−0.485 − 1.93i)29-s + (0.956 − 0.290i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.870 - 0.492i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9942798980\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9942798980\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.773 - 0.634i)T \) |
| 7 | \( 1 + (-0.881 - 0.471i)T \) |
good | 3 | \( 1 + (-0.956 - 0.290i)T^{2} \) |
| 5 | \( 1 + (0.995 + 0.0980i)T^{2} \) |
| 11 | \( 1 + (-1.55 + 0.929i)T + (0.471 - 0.881i)T^{2} \) |
| 13 | \( 1 + (-0.0980 - 0.995i)T^{2} \) |
| 17 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 19 | \( 1 + (-0.634 + 0.773i)T^{2} \) |
| 23 | \( 1 + (0.448 + 0.368i)T + (0.195 + 0.980i)T^{2} \) |
| 29 | \( 1 + (0.485 + 1.93i)T + (-0.881 + 0.471i)T^{2} \) |
| 31 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 37 | \( 1 + (1.61 - 0.577i)T + (0.773 - 0.634i)T^{2} \) |
| 41 | \( 1 + (-0.980 + 0.195i)T^{2} \) |
| 43 | \( 1 + (-0.284 - 1.91i)T + (-0.956 + 0.290i)T^{2} \) |
| 47 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 53 | \( 1 + (-0.289 + 1.15i)T + (-0.881 - 0.471i)T^{2} \) |
| 59 | \( 1 + (-0.0980 + 0.995i)T^{2} \) |
| 61 | \( 1 + (-0.290 + 0.956i)T^{2} \) |
| 67 | \( 1 + (0.174 + 0.235i)T + (-0.290 + 0.956i)T^{2} \) |
| 71 | \( 1 + (-0.482 - 1.59i)T + (-0.831 + 0.555i)T^{2} \) |
| 73 | \( 1 + (-0.555 + 0.831i)T^{2} \) |
| 79 | \( 1 + (-0.162 + 0.108i)T + (0.382 - 0.923i)T^{2} \) |
| 83 | \( 1 + (-0.773 - 0.634i)T^{2} \) |
| 89 | \( 1 + (-0.195 + 0.980i)T^{2} \) |
| 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
−9.539701366314971023218822285767, −8.547418768418727022370730915945, −8.131824273895228791941254137463, −7.26469828120423148770558087720, −6.36955206604365089720224015367, −5.79111218302425120706752004621, −4.71336178450722289812414315735, −3.89738040399808205854146903833, −2.15212037076379916704565136263, −1.26799675683809781506356046889,
1.41040070711846495948599374404, 1.89579926886443591809887114242, 3.73808837823134575534172376315, 4.01874654420244531801499708742, 5.14137654125868567953100891437, 6.66169031054577801291504252553, 7.20194542036143539428746689405, 7.77716979448488955133045330442, 9.022406903763176216666303858101, 9.211222990820565049426595828826