| L(s) = 1 | + (0.707 − 2.12i)5-s + (2 + 2i)7-s − 2.82i·11-s + (−3 + 3i)13-s + (1.41 − 1.41i)17-s − 4i·19-s + (−5.65 − 5.65i)23-s + (−3.99 − 3i)25-s − 9.89·29-s − 8·31-s + (5.65 − 2.82i)35-s + (3 + 3i)37-s − 1.41i·41-s + (8.48 − 8.48i)47-s + i·49-s + ⋯ |
| L(s) = 1 | + (0.316 − 0.948i)5-s + (0.755 + 0.755i)7-s − 0.852i·11-s + (−0.832 + 0.832i)13-s + (0.342 − 0.342i)17-s − 0.917i·19-s + (−1.17 − 1.17i)23-s + (−0.799 − 0.600i)25-s − 1.83·29-s − 1.43·31-s + (0.956 − 0.478i)35-s + (0.493 + 0.493i)37-s − 0.220i·41-s + (1.23 − 1.23i)47-s + 0.142i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.180436503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.180436503\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.707 + 2.12i)T \) |
| good | 7 | \( 1 + (-2 - 2i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.41 + 1.41i)T - 17iT^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + (5.65 + 5.65i)T + 23iT^{2} \) |
| 29 | \( 1 + 9.89T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 43iT^{2} \) |
| 47 | \( 1 + (-8.48 + 8.48i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.07 - 7.07i)T + 53iT^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + (8 + 8i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.65iT - 71T^{2} \) |
| 73 | \( 1 + (-7 + 7i)T - 73iT^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + (8.48 + 8.48i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.41T + 89T^{2} \) |
| 97 | \( 1 + (-3 - 3i)T + 97iT^{2} \) |
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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
−8.740792198871856321937578372876, −7.84100340475059420429563559948, −7.08545569247972139911605637975, −5.92788064036406419057259081347, −5.44396993150058345077812775160, −4.68819201053303965944952136585, −3.87847175613909154757161520135, −2.45266789053355413575472457655, −1.81226128830565253707873920480, −0.34594247557794549912419409904,
1.56689588473057215711326200118, 2.32125957761453669181526416701, 3.60619632621395287329279319980, 4.13187257969429050572756992499, 5.44614954292912893799679770863, 5.78247907050865601705939611314, 7.04532777104796464677301833123, 7.68471014618447380184948510467, 7.78032799268014548221106847945, 9.259710767206533557567446050152
