| L(s) = 1 | + 10.5i·5-s + 8·7-s − 15.8i·11-s + 52.9i·13-s + 14·17-s + 37.0i·19-s − 152·23-s + 12.9·25-s + 158. i·29-s − 224·31-s + 84.6i·35-s + 243. i·37-s + 70·41-s + 439. i·43-s + 336·47-s + ⋯ |
| L(s) = 1 | + 0.946i·5-s + 0.431·7-s − 0.435i·11-s + 1.12i·13-s + 0.199·17-s + 0.447i·19-s − 1.37·23-s + 0.103·25-s + 1.01i·29-s − 1.29·31-s + 0.408i·35-s + 1.08i·37-s + 0.266·41-s + 1.55i·43-s + 1.04·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.444811799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444811799\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 10.5iT - 125T^{2} \) |
| 7 | \( 1 - 8T + 343T^{2} \) |
| 11 | \( 1 + 15.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 52.9iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 14T + 4.91e3T^{2} \) |
| 19 | \( 1 - 37.0iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 152T + 1.21e4T^{2} \) |
| 29 | \( 1 - 158. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 224T + 2.97e4T^{2} \) |
| 37 | \( 1 - 243. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 70T + 6.89e4T^{2} \) |
| 43 | \( 1 - 439. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 336T + 1.03e5T^{2} \) |
| 53 | \( 1 + 31.7iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 534. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 95.2iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 174. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 72T + 3.57e5T^{2} \) |
| 73 | \( 1 + 294T + 3.89e5T^{2} \) |
| 79 | \( 1 - 464T + 4.93e5T^{2} \) |
| 83 | \( 1 + 545. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 266T + 7.04e5T^{2} \) |
| 97 | \( 1 - 994T + 9.12e5T^{2} \) |
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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
−11.50905026458588040193845583089, −10.83158076103272159622626065689, −9.882038112594304649892676144748, −8.799322300174788621536040335678, −7.72570470739018124956570362382, −6.76704944160135269780785338037, −5.80721919713589773528495381596, −4.37683703969393919930731561558, −3.16123559013333408038601033295, −1.71875524449671354734302091813,
0.54128679824374338364716470310, 2.09989410743420151210019375166, 3.84827417093282664065670377297, 5.00570808115154024738561017517, 5.84985551904905678596625356553, 7.39109806090272817479364518718, 8.198969339855747519934614788121, 9.132515518959106249642898717489, 10.11595206648191875139813179680, 11.08809667607036098817194099997
