L(s) = 1 | + (1.08 − 1.95i)5-s + (−0.595 − 2.57i)7-s − 3.74i·11-s + 3.36·13-s − 0.841i·17-s − 5.59i·19-s + 2.35·23-s + (−2.64 − 4.24i)25-s − 1.41i·29-s − 8.66i·31-s + (−5.68 − 1.63i)35-s + 5.15i·37-s + 5.74·41-s + 3.32i·43-s + 6.43i·47-s + ⋯ |
L(s) = 1 | + (0.485 − 0.874i)5-s + (−0.224 − 0.974i)7-s − 1.12i·11-s + 0.931·13-s − 0.204i·17-s − 1.28i·19-s + 0.490·23-s + (−0.529 − 0.848i)25-s − 0.262i·29-s − 1.55i·31-s + (−0.961 − 0.276i)35-s + 0.847i·37-s + 0.896·41-s + 0.507i·43-s + 0.938i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.998089512\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.998089512\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.08 + 1.95i)T \) |
| 7 | \( 1 + (0.595 + 2.57i)T \) |
good | 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 0.841iT - 17T^{2} \) |
| 19 | \( 1 + 5.59iT - 19T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 5.15iT - 37T^{2} \) |
| 41 | \( 1 - 5.74T + 41T^{2} \) |
| 43 | \( 1 - 3.32iT - 43T^{2} \) |
| 47 | \( 1 - 6.43iT - 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 1.82iT - 67T^{2} \) |
| 71 | \( 1 + 3.74iT - 71T^{2} \) |
| 73 | \( 1 + 0.979T + 73T^{2} \) |
| 79 | \( 1 - 6.58T + 79T^{2} \) |
| 83 | \( 1 - 12.5iT - 83T^{2} \) |
| 89 | \( 1 + 2.16T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−8.028301736584860742455249073771, −7.29807890375316423542475317168, −6.28170006170205271724616648854, −5.97197519277242607801303554371, −4.92426348188161405337568375685, −4.30285460164076015266402370583, −3.45429168954028756426123292813, −2.52913523358545318535248104117, −1.17003587381791509459629549535, −0.58091553343580467660412636516,
1.54359893002015869139091226672, 2.23403515177489081165814507265, 3.18303847008051046292793909121, 3.87376965423034126481330851115, 5.01544285655171706228215356863, 5.75728440900851960950906354485, 6.28815576413925818868628679493, 7.04474522749772124847753919657, 7.67668731441690984824822219264, 8.706984378927706013240059339811