| L(s) = 1 | + (1.41 + 1.73i)5-s − 2.82i·7-s − 4.89·11-s + 4.89i·13-s + 3.46i·17-s + 6.92·19-s − 4i·23-s + (−0.999 + 4.89i)25-s − 8.48·29-s − 6.92·31-s + (4.89 − 4.00i)35-s − 4.89i·37-s − 5.65·41-s + 11.3i·43-s + 4i·47-s + ⋯ |
| L(s) = 1 | + (0.632 + 0.774i)5-s − 1.06i·7-s − 1.47·11-s + 1.35i·13-s + 0.840i·17-s + 1.58·19-s − 0.834i·23-s + (−0.199 + 0.979i)25-s − 1.57·29-s − 1.24·31-s + (0.828 − 0.676i)35-s − 0.805i·37-s − 0.883·41-s + 1.72i·43-s + 0.583i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.032325469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.032325469\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.41 - 1.73i)T \) |
| good | 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 - 3.46iT - 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + 4.89iT - 37T^{2} \) |
| 41 | \( 1 + 5.65T + 41T^{2} \) |
| 43 | \( 1 - 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 4.89T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 9.79T + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 9.79iT - 97T^{2} \) |
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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
−9.254213790549855403496790531623, −8.132462538896630160756235529401, −7.30276101490646181205212209766, −7.03624821642363622528630574470, −5.95242326972100168569355506315, −5.31318202514870202669468047529, −4.24926684081838931835722668582, −3.44069468964861175568368768833, −2.44292638938900388123951696750, −1.46969922513456560801788779002,
0.30721985203313656971827043785, 1.77246664072736075136788143594, 2.71030587553550149273665366689, 3.48910775565211473022067299083, 5.17368768080212223334779201587, 5.34847648658898651459571623382, 5.71372043006101662727638982376, 7.20910510145274708493393867634, 7.77245322367120839237532645996, 8.570921853985916740871994006145
