L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.258 − 0.965i)5-s − i·7-s + (−1.36 + 0.366i)10-s − 1.41i·11-s + (0.866 + 0.5i)13-s + (−1.22 + 0.707i)14-s + (0.499 + 0.866i)16-s + (0.707 + 1.22i)17-s − 19-s + (0.707 + 0.707i)20-s + (−1.73 + 1.00i)22-s + (−0.866 − 0.499i)25-s − 1.41i·26-s + ⋯ |
L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.258 − 0.965i)5-s − i·7-s + (−1.36 + 0.366i)10-s − 1.41i·11-s + (0.866 + 0.5i)13-s + (−1.22 + 0.707i)14-s + (0.499 + 0.866i)16-s + (0.707 + 1.22i)17-s − 19-s + (0.707 + 0.707i)20-s + (−1.73 + 1.00i)22-s + (−0.866 − 0.499i)25-s − 1.41i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6704613607\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6704613607\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.258 + 0.965i)T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( 1 + iT - T^{2} \) |
| 41 | \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T^{2} \) |
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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
−10.14673764674272547198365766836, −9.158238639823834985390617231524, −8.581970197951078498728267295972, −7.947551199842191647118411916883, −6.38692211626498296725425356518, −5.63627068413645375342238241766, −4.06879226159713545079683719338, −3.53068470354387277101090052008, −1.87410410066160454750655195747, −0.894409138103094284448744771220,
2.18249335959907441111524844653, 3.33945659040855906836525519565, 5.03225451643574385288856006494, 5.85393472339357701418459760224, 6.65468675774850185327414129291, 7.31055820282571837562642379911, 8.127858544296409656725640608433, 9.023325534312118953649715038393, 9.714263529502166216308532738357, 10.42962977440213615866889040976