| L(s) = 1 | − 2.44·5-s + 3.46i·7-s + 2.82i·11-s + 3.46i·13-s + 1.41i·17-s + 4·19-s − 4.89·23-s + 0.999·25-s + 2.44·29-s − 3.46i·31-s − 8.48i·35-s + 1.41i·41-s − 8·43-s + 4.89·47-s − 4.99·49-s + ⋯ |
| L(s) = 1 | − 1.09·5-s + 1.30i·7-s + 0.852i·11-s + 0.960i·13-s + 0.342i·17-s + 0.917·19-s − 1.02·23-s + 0.199·25-s + 0.454·29-s − 0.622i·31-s − 1.43i·35-s + 0.220i·41-s − 1.21·43-s + 0.714·47-s − 0.714·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0917 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.601038 + 0.658963i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.601038 + 0.658963i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 - 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 - 2.44T + 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 4.89T + 47T^{2} \) |
| 53 | \( 1 - 7.34T + 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 13.8iT - 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 14.6T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 14.1iT - 83T^{2} \) |
| 89 | \( 1 + 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−11.84015567944843766573212915672, −11.59667052496761948441829914566, −10.08859469931471545410726790855, −9.154311908649194061026174889255, −8.227849519968208422563471102326, −7.31423318661019843731643420217, −6.12341103912649093326759272255, −4.86337652933367729055055319932, −3.74312274981843969306408564275, −2.17511110363757104795173775784,
0.68380938467871172893496655527, 3.25484984870940174907580217689, 4.08343408926785093362318250365, 5.42307176504799492297468197603, 6.87351761499343924902840315998, 7.74768226398637889560513790161, 8.397330527421973118006201571349, 9.882823994756960579045756651823, 10.69117842925833683616591519721, 11.51879985194321367563687121048
