L(s) = 1 | − 3.14·3-s + 6.89·9-s − 6.61i·11-s − 7.89i·17-s − 2.51i·19-s − 12.2·27-s + 20.7i·33-s + 12.7·41-s − 8.48·43-s + 7·49-s + 24.8i·51-s + 7.89i·57-s − 14.1i·59-s + 7.88·67-s + 13.6i·73-s + ⋯ |
L(s) = 1 | − 1.81·3-s + 2.29·9-s − 1.99i·11-s − 1.91i·17-s − 0.575i·19-s − 2.36·27-s + 3.62i·33-s + 1.99·41-s − 1.29·43-s + 49-s + 3.48i·51-s + 1.04i·57-s − 1.84i·59-s + 0.962·67-s + 1.60i·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6643328976\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6643328976\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 3.14T + 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 6.61iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.89iT - 17T^{2} \) |
| 19 | \( 1 + 2.51iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 7.88T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.6iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 14.1T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.308935960710054602172418091074, −7.33196952513675295407161116547, −6.68751795129918630418002375728, −5.98630894364181897922897366852, −5.36304032863979252518327431293, −4.80706392265289957854772789464, −3.74429926084785172068912350875, −2.66504344399960320604746914705, −0.964447248041121248330458733957, −0.35822190437345616787880044762,
1.27424355067160224852716213942, 2.09856075260930554006508240854, 3.94916117737574754190428489502, 4.41440218973449295562084928720, 5.25861125753129682366104801408, 5.97768648638132833399650763546, 6.58041521312045408377561461079, 7.30812792612669028754917165941, 7.976036381197546904797758051111, 9.202037167888371682219379941573