L(s) = 1 | + 3.04i·5-s + 5.27i·7-s + 3·9-s + 6.50·11-s − 7.27·17-s + 4.35·19-s − 4i·23-s − 4.27·25-s − 16.0·35-s + 5.67·43-s + 9.13i·45-s − 2.72i·47-s − 20.8·49-s + 19.8i·55-s − 10.8i·61-s + ⋯ |
L(s) = 1 | + 1.36i·5-s + 1.99i·7-s + 9-s + 1.96·11-s − 1.76·17-s + 1.00·19-s − 0.834i·23-s − 0.854·25-s − 2.71·35-s + 0.865·43-s + 1.36i·45-s − 0.397i·47-s − 2.97·49-s + 2.67i·55-s − 1.38i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.904690571\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.904690571\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35T \) |
good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 3.04iT - 5T^{2} \) |
| 7 | \( 1 - 5.27iT - 7T^{2} \) |
| 11 | \( 1 - 6.50T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 5.67T + 43T^{2} \) |
| 47 | \( 1 + 2.72iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 10.8iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 5.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 8.71T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.775214285472108847226613744250, −9.196801548246484707035156648990, −8.574628282386600801058380621169, −7.25452009210317472523627177231, −6.52467784972191794386145344525, −6.15209206532998145784463642928, −4.81343188185027416642311696102, −3.76330034538093440298926728770, −2.68079062776598013186740830543, −1.82195675813168942972789442970,
0.944978688094843935553310537779, 1.48741051580411768737481763335, 3.81443903747514262171292861300, 4.19340999346431843366418750335, 4.87913046191056776700861523970, 6.37353295896070616251461915065, 7.07244341123600810908454005073, 7.70388633348990962944087145668, 8.942535280941519484153511228276, 9.365405257597178219175939964424