L(s) = 1 | + 3.64i·2-s + (1.72 − 8.83i)3-s + 2.68·4-s − 34.1i·5-s + (32.2 + 6.30i)6-s + 45.8·7-s + 68.1i·8-s + (−75.0 − 30.5i)9-s + 124.·10-s + 36.4i·11-s + (4.64 − 23.7i)12-s + 161.·13-s + 167. i·14-s + (−301. − 59.0i)15-s − 205.·16-s − 35.8i·17-s + ⋯ |
L(s) = 1 | + 0.912i·2-s + (0.191 − 0.981i)3-s + 0.167·4-s − 1.36i·5-s + (0.895 + 0.175i)6-s + 0.934·7-s + 1.06i·8-s + (−0.926 − 0.376i)9-s + 1.24·10-s + 0.301i·11-s + (0.0322 − 0.164i)12-s + 0.952·13-s + 0.852i·14-s + (−1.34 − 0.262i)15-s − 0.803·16-s − 0.124i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.981 + 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.69984 - 0.164632i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.69984 - 0.164632i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.72 + 8.83i)T \) |
| 11 | \( 1 - 36.4iT \) |
good | 2 | \( 1 - 3.64iT - 16T^{2} \) |
| 5 | \( 1 + 34.1iT - 625T^{2} \) |
| 7 | \( 1 - 45.8T + 2.40e3T^{2} \) |
| 13 | \( 1 - 161.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 35.8iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 549.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 363. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 969. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.81e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.59e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 917. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 542.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.64e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.06e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 4.25e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.62e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 3.87e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.89e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.85e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 7.17e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 8.35e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.54e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.08e4T + 8.85e7T^{2} \) |
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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
−15.99664947522209879193986399462, −14.77277016793859423438334625846, −13.60031978430914522665229451375, −12.41843410149773512966153919144, −11.25431202834835792618968726775, −8.644429458060921778483534589698, −8.097124965471871411988057231225, −6.52528552088797013105789927597, −5.05722884183438551097383424165, −1.59853907764022310733962055026,
2.52729682595026094871170167433, 4.01121938847311481563989521074, 6.40454546369455813091916543456, 8.408012983159543007362565168062, 10.29915008318851130575457176464, 10.81442999955468437146287575402, 11.71475974714574010973290517059, 13.73632397268419881857056930356, 14.89382986596939026790408599174, 15.67066297753761982678682690492